Differentiability: Corners, Cusps & Continuity

The Definition of Differentiability

The derivative formula can always be written down, but it does not always give a number.

The rules of differentiation hand out a slope at every point, as if every graph had one. Most do: at

x = 1

the curve

y = x^2

has exactly one tangent line. But at

x = 0

the graph of

y = |x|

comes in at one angle and leaves at another, and "the slope there" has no single answer. Something has to say in advance which points get a derivative and which do not.

Left:

y = x^2

and its one tangent at

(1, 1)

. Right:

y = |x|

, whose two dashed directions at

(0, 0)

disagree.

The definition

A function

f

is differentiable at

x = a

if the limit below exists and is finite:

The Definition of Differentiability

f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}

Each dashed secant joins

(1, 1)

to a point

h

further along

y = x^2

; its slope is the difference quotient. As

h

shrinks the secants settle on the solid tangent, slope

2

— the limit in the box.

This is a two-sided limit: the difference quotient must approach the same finite value whether

h \to 0^+

or

h \to 0^-

.

Differentiability is a property of a function at a point: it either holds there or it does not.

It is a yes-or-no question asked of one function at one point. When the answer is yes, the limit is a number — the derivative

f'(a)

, the slope of the tangent. So

x^2

is differentiable at

1

with

f'(1) = 2

; the same function at another point is a separate question with its own answer.

On numbers, the difference quotient of x² at 1 closes in on 2 from both sides.

For

f(x) = x^2

at

a = 1

, the quotient is

\frac{(1 + h)^2 - 1}{h}

. Take

h

smaller and smaller, positive and negative.

h	(1 + h)²	difference quotient	equals 2 + h
0.1	1.21	0.21 / 0.1	2.1
0.01	1.0201	0.0201 / 0.01	2.01
0.001	1.002001	0.002001 / 0.001	2.001
−0.001	0.998001	−0.001999 / −0.001	1.999
−0.01	0.9801	−0.0199 / −0.01	1.99
−0.1	0.81	−0.19 / −0.1	1.9

The last column is always

2 + h

, since

(1 + h)^2 - 1 = 2h + h^2

. Positive

h

approaches

2

from above, negative

h

from below, and both sides agree: the limit exists, it is finite, and

f'(1) = 2

.

The definition keeps its promises.

It promised a tangent line, and there is one: through

(1, 1)

with slope

2

, the line

y = 2x - 1

, which touches

y = x^2

only at that point, because

x^2 - (2x - 1) = (x - 1)^2 \geq 0

. It also promised that a differentiable function cannot jump: as

h \to 0

the top of the quotient,

f(1 + h) - f(1)

, must go to

0

for the quotient to stay finite, so

f(1 + h) \to f(1)

. Being differentiable forces being continuous.

The converse fails: a continuous function can still lack a derivative.

Run the same test on

|x|

at

0

. The quotient is

\frac{|h|}{h}

, and it never settles on one number.

h	\|h\| / h
0.1, 0.01, 0.001, …	1
−0.1, −0.01, −0.001, …	−1

Each side has its own limit,

1

and

-1

, but they disagree, so the two-sided limit does not exist — even though

|x|

has no break at

0

. If the limit does not exist, or if it equals

\pm\infty

, or if

f(a)

itself is undefined, then

f

is not differentiable at

a

.

The rest of the page goes through the notation, why differentiability implies continuity, each way a derivative can fail — corners, cusps, vertical tangents and discontinuities — one-sided derivatives, differentiability on intervals, and pathological examples.

Differentiability Notation

Notation

Differentiability Notation

The one-sided derivative marks, the infinite-slope verdict, and what “differentiable on

[a,b]

” actually promises.

The base

f'

and the difference quotient — the derivative function; the superscript directions on limits — one-sided limits.

f'_{-}(a)

·

f'_{+}(a)

f prime minus at a; f prime plus at a

Left- and right-hand derivatives — the difference quotient limited from one side, defined in One-Sided Derivatives below. Watch where the sign sits: subscript on the prime, not superscript on the point — here the direction belongs to the operation, not to

a

.

CasesDifferentiable at

a

iff both exist, finite, and equal. Endpoints of

[a,b]

get one apiece. Piecewise boundaries need two checks: continuity, then

f'_{-} = f'_{+}

.

Also written

f'(a^{-})

,

f'(a^{+})

— the superscript variant, riskier because of the collision below.

Do not confuse

\lim_{x \to a^{+}} f'(x)

— the one-sided limit of the derivative, a different object. For

x^2\sin(1/x)

at

0

the right-hand derivative exists (it is

0

) while the limit of

f'

does not — see Continuity of the Derivative.

f'(a) = \pm\infty

The derivative is infinite

The equals-sign abuse imported from infinite limits: a verdict, not a value. Differentiability demands a finite limit, so this equation announces failure — but the cleanest kind, a vertical tangent.

CasesBoth sides infinite, same sign — vertical tangent, Vertical Tangents below. Opposite signs — cusp. Finite but unequal — corner: no infinity involved, the failure is disagreement.

Also written

|f'(x)| \to \infty

as

x \to a

— the form careful texts use to avoid the abuse.

Do not confuseA derivative that “exists and is infinite”. In the definition, existence means finite —

= \infty

is one of the ways existence fails, not a value of it.

f

differentiable on

[a,b]

f is differentiable on the closed interval from a to b

A convention, not just a quantifier: two-sided derivatives throughout

(a,b)

, plus

f'_{+}(a)

and

f'_{-}(b)

at the ends — mirroring continuity on a closed interval.

CasesOpen

(a,b)

— two-sided everywhere, no endpoint clause. The closed-interval convention is what lets Rolle and the Mean Value Theorem state their hypotheses on

[a,b]

at all.

Also written

f \in C^1[a,b]

when

f'

is moreover continuous — one rung up the smoothness ladder.

Do not confuseTwo-sided differentiability at every point of

[a,b]

— impossible at the endpoints, where one side of the quotient has nothing to stand on.

All calculus symbols →↑ Derivatives

Differentiability Implies Continuity

If

f

is differentiable at

a

, then

f

is continuous at

a

. The proof is direct.

Differentiability Implies Continuity

f \text{ differentiable at } a \implies f \text{ continuous at } a

Learn more about this formula: Differentiability Implies Continuity →

Browse all derivative formulas →

Rewrite

f(x) - f(a)

as

Differentiability Implies Continuity

f(x) - f(a) = \frac{f(x) - f(a)}{x - a} \cdot (x - a)

Fact: Differentiable at one point only

Let

f(x) = x^2

for rational

x

and

f(x) = 0

for irrational

x

. Its difference quotient at

0

is at most

|h|

in size, so

f'(0) = 0

, yet

f

is not continuous at any other point.

As

x \to a

, the first factor approaches

f'(a)

(which exists by assumption) and the second factor approaches

0

. Their product approaches

f'(a) \cdot 0 = 0

, so

\lim_{x \to a} f(x) = f(a)

.

The argument is local. It starts from the derivative at the single point

a

and ends with continuity at that same point, and at no step does it look at the values of

f

at points near

a

. A derivative at one point is a statement about that point alone.

This establishes a hierarchy: differentiability is a stronger condition than continuity. Every differentiable function is continuous, but the reverse implication fails. The functions that are continuous but not differentiable at a point form a significant and instructive class.

Corners

A corner occurs when the left-hand and right-hand derivatives both exist as finite numbers but disagree.

Corners

\lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} \neq \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}

The graph makes a sharp turn at

x = a

—two distinct tangent directions meeting at a single point. The function is continuous at

a

(no break in the graph), but the two-sided limit of the difference quotient fails to exist because the one-sided limits differ.

Pitfall: Assuming every absolute value makes a corner

Seeing

|x|

in

f(x) = x|x|

and declaring a corner at

0

skips the computation. The quotient is

\frac{h|h|}{h} = |h|

, which tends to

0

from both sides, so

f'(0) = 0

and there is no corner.

Cusps

A cusp occurs when both one-sided derivatives are infinite but with opposite signs. As

h \to 0^+

, the difference quotient approaches

+\infty

(or

-\infty

), and as

h \to 0^-

, it approaches the opposite.

The graph comes to a sharp point where the curve doubles back on itself with vertical tangent directions from both sides. Unlike a corner, the slopes do not stay finite—they blow up.

Pitfall: Checking only one side

For

f(x) = x^{4/5}

at

0

, checking only

h > 0

gives the quotient

h^{-1/5} \to +\infty

and suggests a vertical tangent. For

h < 0

the quotient is negative and tends to

-\infty

, so the point is a cusp.

Vertical Tangents

A vertical tangent occurs when the two-sided limit of the difference quotient exists but equals

+\infty

or

-\infty

. Both sides agree on the direction—unlike a cusp—but the slope is infinite.

The standard example is

f(x) = x^{1/3}

at

x = 0

. The derivative formula

f'(x) = \frac{1}{3}x^{-2/3}

gives

f'(x) \to +\infty

as

x \to 0

from either side. The graph passes smoothly through the origin with no sharp point, but the tangent line there is the vertical line

x = 0

.

A vertical tangent line has undefined slope, so the derivative does not exist in the usual sense. The function is continuous and the graph is smooth, yet the rate of change is unbounded at that point. This is the mildest form of non-differentiability—the geometry is clean, but the numerical value of the derivative is not finite.

A smooth curve with a vertical tangent

The section’s x^(1/3) at x = 0. The curve passes through the origin without a sharp point and stands vertical there: the slope grows to +∞ from both sides, and the tangent is the dashed line x = 0. The tangent exists as a line, but its slope is not a number, so the function is not differentiable at 0.

Compare section 4: here both sides agree on +∞, so the curve has no sharp point, only a vertical tangent.

Discontinuities

COROLLARY

If

f

is not continuous at

a

, then

f

is not differentiable at

a

.

Since differentiability implies continuity, any discontinuity automatically rules out differentiability.

At a jump discontinuity, the function leaps from one value to another. No tangent line can bridge the gap. At an infinite discontinuity (vertical asymptote), the function blows up—even if

f(a)

is assigned a value, the graph near

a

does not approach the point

(a, f(a))

, so no tangent can pass through it. At a removable discontinuity, the limit exists but either

f(a)

is undefined or differs from the limit; redefining

f(a)

to equal the limit restores continuity and may restore differentiability.

Discontinuities are the most straightforward obstruction to differentiability. The other failure modes—corners, cusps, vertical tangents—are subtler because the function is continuous at the problematic point.

Jump at x = 0: the two sides disagree

Approaching zero from the left the function heads to 0, from the right it sits at 1, and the checker’s three conditions fail at the second: no two-sided limit, so no continuity, so no derivative. A tangent line would have to be two different lines at once. Test any breakpoint against the three conditions on the continuity checker.

Continuity is thus the first gate a function must pass before differentiability can even be asked about.

One-Sided Derivatives

The left-hand derivative of

f

at

a

is

One-Sided Derivatives

f'_-(a) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h}

and the right-hand derivative is

One-Sided Derivatives

f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}

One-Sided Derivative

f'_-(a) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} \qquad f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}

Learn more about this formula: One-Sided Derivative →

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THEOREM

The two-sided derivative

f'(a)

exists if and only if both one-sided derivatives exist, are finite, and are equal:

f'_-(a) = f'_+(a)

.

Pitfall: Differentiating the pieces instead of using f(a)

For

f(x) = x^2

on

x \leq 1

and

2x + 1

on

x > 1

, the piece formulas give slopes

2

and

2

at

x = 1

. But

f(1) = 1

while the right piece tends to

3

, so

f'_+(1) = \lim_{h \to 0^+} \frac{2 + 2h}{h} = +\infty

and

f

is not differentiable at

1

.

Derivative	Notation	Defining limit	Where it is the natural object
Left-hand	f'₋(a)	lim_h→0⁻ [f(a+h) − f(a)] / h	right endpoint of an interval; left side of a piecewise boundary
Right-hand	f'₊(a)	lim_h→0⁺ [f(a+h) − f(a)] / h	left endpoint of an interval; right side of a piecewise boundary
Two-sided	f'(a)	requires f'₋(a) = f'₊(a), both finite	interior points; the standard derivative

Differentiability on Intervals

A function is differentiable on an open interval

(a, b)

if it is differentiable at every point in

(a, b)

. No endpoint conditions are needed because open intervals exclude their endpoints.

On a closed interval

[a, b]

, the convention is:

f

is differentiable on

(a, b)

and possesses a right-hand derivative at

a

and a left-hand derivative at

b

. This mirrors the definition of continuity on a closed interval and ensures that derivative-based theorems—the Mean Value Theorem, Rolle's Theorem—apply on

[a, b]

.

For piecewise functions defined on adjacent intervals, differentiability at each interior boundary requires two conditions: continuity at the boundary (the pieces connect) and agreement of the one-sided derivatives (the pieces connect smoothly). Continuity alone is not sufficient—

f(x) = |x|

is continuous at

x = 0

but not differentiable there.

Setting	Differentiable means	Notes
Open interval (a, b)	f differentiable at every point in (a, b)	no endpoint conditions — endpoints are excluded
Closed interval [a, b]	differentiable on (a, b); right-hand derivative at a; left-hand derivative at b	required hypothesis for MVT and Rolle's Theorem
Piecewise boundary at a	f continuous at a AND f'₋(a) = f'₊(a) (both finite)	continuity alone is not enough — \|x\| is continuous at 0 but not differentiable

Two pieces join differentiably only when they meet and their slopes agree at the join.

Pathological Examples

The gap between continuity and differentiability extends further than isolated points. The Weierstrass function, constructed in 1872, is continuous at every real number but differentiable at none. Its graph is an infinitely jagged curve with no smooth segments anywhere.

Fact: A tangent anywhere is the exception

In 1931 Banach and Mazurkiewicz proved that, in the sense of Baire category, a typical continuous function on

[0, 1]

has a derivative at no point. The reason: zigzags with huge slopes can be added to any continuous graph while moving it as little as you like.

This is not an isolated curiosity. Entire families of continuous-but-nowhere-differentiable functions exist, and they arise naturally in fractal geometry and stochastic processes. Brownian motion paths, for instance, are almost surely continuous everywhere and differentiable nowhere.

These examples demonstrate that differentiability is a genuinely restrictive condition. Continuity guarantees an unbroken graph; differentiability demands a smooth one. Most "natural" functions in calculus—polynomials, trigonometric functions, exponentials—are differentiable on their entire domains, which can create the false impression that differentiability is routine. The pathological cases reveal that smooth behavior is special, not default.

Summary: A Typology of Differentiability Failure

The failure modes covered above—corners, cusps, vertical tangents, and discontinuities—each block differentiability through a different mechanism and leave a different signature on the graph. The table below collects them in parallel, organized by what the one-sided derivatives do, whether the function remains continuous at the problem point, and a canonical example for each. Reading across a row identifies the mode; reading down a column reveals what changes from one mode to the next.

Failure mode	Left-hand f'(a)	Right-hand f'(a)	Continuous at a?	Canonical example
Corner	finite	finite, different from left	yes	f(x) = \|x\| at x = 0 (slopes −1 and +1)
Cusp	±∞	∓∞ (opposite sign)	yes	f(x) = x^2/3 at x = 0
Vertical tangent	±∞	±∞ (same sign as left)	yes	f(x) = x^1/3 at x = 0
Discontinuity	undefined or DNE	undefined or DNE	no	step function at the jump; 1/x at 0

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Before you read

Four ways a derivative can fail

A function can fail to be differentiable at a point in four distinct ways. Three of them keep the function continuous.

Differentiability failure4 types

Corner

|x|

at

0

One-sided derivativesfinite but unequal

Continuousyes

Cusp

x^{2/3}

at

0

One-sided derivativesinfinite, opposite signs

Continuousyes

Vertical tangent

x^{1/3}

at

0

Derivative limit

\pm\infty

from both sides

Continuousyes, smooth graph

Discontinuity

Kindsjump, infinite or removable

not continuous, so not differentiable

continuous but not differentiablenot even continuous

Takeaway

Continuity is not enough

A corner, a cusp or a vertical tangent can sit on a continuous graph. Only a discontinuity rules out differentiability before any derivative is computed.

Four ways differentiability fails·/calculus/derivatives/differentiabilityLearn Math Class

Common Pitfalls

Four mistakes cause most wrong verdicts about differentiability. Three of them are marked where they happen in the sections above; the first is the one the sections on continuity and on intervals warn against. Here each is worked through in full, with the wrong and the right working side by side.

Assuming continuity guarantees a derivative

Because differentiability implies continuity, the converse is assumed as well: a function whose graph has no break at a point is taken to have a derivative there.

Wrong working:

f(x) = |x - 2|

is continuous at

x = 2

, so

f'(2)

exists, and by symmetry

f'(2) = 0

.

Right working: the difference quotient is

\frac{f(2 + h) - f(2)}{h} = \frac{|h|}{h}

, which is

1

for

h > 0

and

-1

for

h < 0

. The one-sided derivatives

-1

and

1

disagree, so

f'(2)

does not exist.

Why it happens: the implication is learned as a link between the two words, and the direction gets lost. An unbroken graph also looks well behaved enough to have a slope.

How to check: compute the quotient for one small positive and one small negative

h

. At

h = 0.01

it is

1

, at

h = -0.01

it is

-1

.

Assuming every absolute value makes a corner

An absolute value in the formula is taken as proof of a corner, and the function is declared non-differentiable at the point where the inside is

0

, without computing anything.

Wrong working:

f(x) = x|x|

contains

|x|

, which has a corner at

0

, so

f

has a corner at

0

and

f'(0)

does not exist.

Right working:

f(0) = 0

, and the difference quotient is

\frac{h|h|}{h} = |h|

, which tends to

0

from both sides. So

f'(0) = 0

. Piece by piece,

f(x) = x^2

for

x \geq 0

and

f(x) = -x^2

for

x < 0

, and both have slope

0

at

0

.

Why it happens: the corner of

|x|

is remembered as a property of the symbol rather than as the result of a computation. The extra factor

x

flattens the graph near

0

and removes the corner.

How to check: evaluate the quotient. At

h = 0.01

and at

h = -0.01

it is

0.01

, small on both sides and shrinking with

h

.

Checking only one side

To tell a cusp from a vertical tangent, the slope is examined for positive

h

(or positive

x

) only, and the sign on the other side is never checked.

Wrong working: for

f(x) = x^{4/5}

,

f'(x) = \frac{4}{5}x^{-1/5}

grows without bound as

x \to 0^+

, so

f'(x) \to +\infty

and the graph has a vertical tangent at

0

.

Right working: the quotient is

\frac{f(h) - f(0)}{h} = \frac{h^{4/5}}{h} = h^{-1/5}

. For

h > 0

it tends to

+\infty

. For

h < 0

,

h^{1/5}

is negative, so

h^{-1/5}

is negative and tends to

-\infty

. The signs are opposite, so

0

is a cusp, as for

x^{2/3}

.

Why it happens: small positive numbers are the default test values, and the formula

x^{-1/5}

does not show its sign for negative

x

at a glance.

How to check: try one value on each side. At

h = 10^{-5}

the quotient is

10

; at

h = -10^{-5}

it is

-10

. The values of

f

give a second check:

x^{4/5} \geq 0

on both sides of

0

, so the graph comes down to the origin and goes back up, which is the shape of a cusp; a vertical tangent such as

x^{1/3}

passes from negative to positive values.

Differentiating the pieces instead of using f(a)

At the boundary of a piecewise function, each piece's formula is differentiated and the two slopes are compared, without checking that the pieces meet.

Wrong working: for

f(x) = x^2

on

x \leq 1

and

f(x) = 2x + 1

on

x > 1

, the left formula has derivative

2x

, which is

2

at

x = 1

, and the right formula has derivative

2

. The slopes match, so

f'(1) = 2

.

Right working: first check continuity.

f(1) = 1

from the left piece, but the right piece tends to

2 \cdot 1 + 1 = 3

, so

f

jumps at

1

. The right-hand derivative uses

f(1)

:

\lim_{h \to 0^+} \frac{2(1 + h) + 1 - 1}{h} = \lim_{h \to 0^+} \frac{2 + 2h}{h} = +\infty

. So

f

is not differentiable at

1

; it is not even continuous there.

Why it happens: the derivative of a piece's formula is the slope of that piece's own graph. It knows nothing about the value

f(1)

or about whether the two graphs touch.

How to check: before comparing slopes, put the boundary value into both formulas. Here they give

1

and

3

, which settles the question. The quotient confirms it: at

h = 0.01

the right-hand quotient is

202

.

Differentiability FAQ

Can a derivative be infinite?

+

No. In the definition, existence means the limit is finite, so writing f′(a) = ±∞ records a failure rather than a value — a borrowed abuse of the equals sign from infinite limits. Careful texts write |f′(x)| → ∞ instead. Geometrically this is the mildest failure: a tangent line still exists, it is just vertical, so it has no slope.Read more →

Is a cusp the same as a corner?

+

No. Both are sharp points on a continuous graph, but the one-sided derivatives behave differently. At a corner they are finite and unequal — |x| at 0 gives −1 and +1. At a cusp they are both infinite with opposite signs, as with x^(2/3) at 0, so the curve doubles back on itself. A cusp is the sharper of the two.Read more →

How do you check whether a piecewise function is differentiable at a boundary?

+

Two conditions, in order. First check continuity at the boundary — the two pieces must agree on the value there, or differentiability is impossible. Then compute the one-sided derivatives from each piece separately and check that they are equal and finite. Continuity alone is not enough: |x| is continuous at 0 but its one-sided derivatives are −1 and +1.Read more →

Is there a function that is continuous everywhere but differentiable nowhere?

+

Yes. The Weierstrass function, constructed in 1872, is continuous at every real number and differentiable at none — its graph is jagged at every scale with no smooth segment anywhere. Such functions are not isolated curiosities: they appear throughout fractal geometry, and the paths of Brownian motion are almost surely continuous everywhere and differentiable nowhere.Read more →

Does the one-sided derivative equal the one-sided limit of f′?

+

Not necessarily — they are different objects. The right-hand derivative limits the difference quotient at the single point a, while the one-sided limit of f′ asks what the derivative function approaches near a. For x²sin(1/x) at 0 the right-hand derivative exists and equals 0, yet f′ has no limit there because it oscillates.Read more →

Differentiability

When the Derivative Exists — and When It Does Not

Key Terms

The Definition of Differentiability

Differentiability Notation

Differentiability Notation

Differentiability Implies Continuity

Corners

Cusps

Vertical Tangents

Discontinuities

One-Sided Derivatives

Differentiability on Intervals

Pathological Examples

Summary: A Typology of Differentiability Failure

Common Pitfalls

Differentiability FAQ

Can a derivative be infinite?

Is a cusp the same as a corner?

How do you check whether a piecewise function is differentiable at a boundary?

Is there a function that is continuous everywhere but differentiable nowhere?

Does the one-sided derivative equal the one-sided limit of f′?