The Function picker on the left lists six families, from Smooth (x²) to Staircase; tapping one loads it and parks c on its first point of interest, or at 0 for Smooth (x²). Learn more about the function families
The Display checkboxes f(x), L⁻, L⁺ and x = c each hide or show one layer, and the legend under the plot lists only the layers that are on; Accent color under Appearance recolours the highlights. Learn more about the display toggles
The point c slider under Parameters moves the probe from −3 to 3 in steps of 0.05, and the checklist is re-run at every position. Learn more about the c slider
The Jump to buttons under the slider put c exactly on the family's points of interest, one for most families and five for Staircase; Reset returns c to the first of them. Learn more about the jump-to buttons
The plot draws f as a solid curve, the one-sided limits as dashed horizontal lines with markers at (c,L−) and (c,L+), and x=c as a dashed vertical line that follows the slider. Learn more about probing with the c slider
The Three-condition checklist marks each condition ✓ or ✕, f(c) is defined, the limit exists, f(c) equals the limit, and its last row tags the point continuous, removable, jump or infinite. Learn more about the three-condition checklist
Continuity at a point — the function f is continuous at c when three conditions all hold: f(c) is defined, limx→cf(x) exists, and the two are equal.
One-sided limits — the left limitf(c−)=limx→c−f(x) and the right limit f(c+)=limx→c+f(x). The two-sided limit exists exactly when both one-sided limits exist and agree.
Removable discontinuity — a point where the two-sided limit exists but either f(c) is undefined or f(c) differs from the limit. Redefining f(c) to the limit value repairs the function.
Jump discontinuity — a point where both one-sided limits are finite but unequal. No single value of f(c) can repair it.
Continuous on an interval — f is continuous at every point in the interval.
Getting Started
The page opens with the Hole at x = 1 family pre-loaded, with the probing point c already parked at the discontinuity. The three-condition checklist below the graph shows two passes and one fail at c=1, classifying the point as a removable discontinuity.
To explore quickly:
• Pick a different function family from the left panel — Smooth, Hole, Jump, Wrong value, Asymptote, or Staircase. Each demonstrates a specific way continuity can pass or fail.
• Drag the point c slider to move the probing position through the graph. The dashed vertical line on the plot follows c, and the three checklist rows update live.
• Use the Jump to buttons (when present) to land directly on the family's known points of interest — the few values of c where the checklist actually has something to fail.
• The verdict box at the bottom of the checklist gives the one-word classification: continuous, removable, jump, or infinite.
The Function Families
DemoPicking a family
Step 0 of 5
Each family in the left panel is chosen to isolate one continuity behavior:
• Smooth (x²) — a polynomial, continuous everywhere. The checklist passes at every c. Useful as a baseline.
• Hole at x = 1 — the rational function (x2−1)/(x−1), which simplifies algebraically to x+1 but is undefined at x=1. The limit exists; the value does not.
• Jump at x = 0 — a piecewise function whose two pieces produce different one-sided limits at the seam. The two-sided limit fails to exist.
• Wrong value at x = 1 — the line x+1 everywhere except f(1)=0. The limit exists, the value exists, but they disagree.
• Asymptote at x = 0 — the function 1/x2, with both one-sided limits going to +∞. Not removable.
• Staircase — ⌊x⌋+0.5. A jump at every integer; continuous between them.
Smooth: All Three Conditions Pass
The Smooth family is f(x)=x2, a polynomial, and polynomials are continuous on the whole real line. There is no point of interest to jump to because there is nothing to find — the checklist passes at every c you can slide to.
At c=0 the readings are f(c)=0, L−=0, L+=0. All three rows turn green: the value exists, the two one-sided limits agree, and their common value matches f(c).
Smooth, frozen at c = 0
f(x) = x² with the probe at the origin. One L line, one marker, nothing else to see - f(c), L⁻ and L⁺ all read 0 and the three checks pass.
This is the baseline the other five families are measured against, and it is worth spending a moment on before moving to the failures. Continuity is the ordinary case — the interesting question is never "why is this function continuous here" but "what exactly broke at the one point where it isn't".
Slide c anywhere on this family and watch nothing happen. The dashed vertical moves, the L line follows it, the three checks stay green.
Hole: the Value Is Missing
The Hole family is f(x)=x−1x2−1. Factoring gives x−1(x−1)(x+1), which cancels to x+1 — but only for x=1. At x=1 the original expression is 00, so the function simply has no value there.
At the point of interest c=1: L−=2 and L+=2, so condition 2 passes cleanly. Condition 1 fails, because f(1) is undefined. The verdict is a removable discontinuity.
Hole, frozen at c = 1
The L line sits at 2 and the curve passes straight through that level, but no f(c) marker sits on it: the function has no value at x = 1. Condition 1 is the one that fails.
Removable is the right word in the strict sense: define f(1)=2 and every condition passes. The repair is a single point, and the limit tells you exactly which value to use.
Note what the graph does and does not show. The curve looks like the line y=x+1 with a gap you would miss at a glance, which is the practical danger of this case — the algebra says the point is absent, not the picture.
Jump: the One-Sided Limits Disagree
The Jump family is piecewise: f(x)=x for x<0, and f(x)=x+1 for x≥0. Each piece is continuous on its own; the trouble is at the seam.
At c=0: L−=0 approaching from the left, L+=1 approaching from the right. Two different finite values, so the two-sided limit does not exist and condition 2 fails. Condition 1 passes — f(0)=1 is perfectly well defined.
Jump, frozen at c = 0
Two L lines instead of one, at 0 and at 1. The closed endpoint belongs to the upper piece and the open one to the lower - the visual signature of one-sided limits that disagree.
This one is not repairable, and the reason is worth stating precisely. A patch can only change f(c), a single number. Here no single number equals both 0 and 1, so no redefinition of f(0) can make the limit exist. The gap has size L+−L−=1, and that gap is a property of the two pieces, not of the value at the seam.
That is the whole distinction between a removable discontinuity and a jump: removable means the limit exists and the value is wrong or missing; jump means the limit itself fails, and the types of discontinuities section works through the full classification.
Wrong Value: the Limit Exists but Misses
The Wrong value family is the line f(x)=x+1 everywhere except at a single point, where the definition overrides it: f(1)=0 instead of the natural 2.
At c=1 the first two conditions pass — f(1)=0 is defined, and L−=L+=2 so the limit exists. It is condition 3 that fails: f(c)=limx→cf(x), because 0=2. The graph shows it plainly, with the isolated dot sitting well below the line it belongs to.
Wrong value, frozen at c = 1
The L line is at 2 where the line would go, while the f(c) marker sits alone at 0. Both are drawn, and the vertical distance between them is the failure of condition 3.
The verdict is again removable, and this family exists to make the point that removable covers two different failures. In the hole family condition 1 failed and the value was missing; here condition 1 passes and condition 3 fails, because the value is present but wrong. Both are repaired the same way — set f(1)=2, the value the limit dictates.
That shared repair is why the classification groups them together. What matters for repairability is whether limx→cf(x) exists, not whether f(c) happens to be defined.
Asymptote: an Infinite Discontinuity
The Asymptote family is f(x)=x21, which grows without bound on both sides of the origin.
At c=0 the tool reports f(c) as undefined and both one-sided limits as +∞. Condition 1 fails and condition 2 fails as well — and it fails for a stronger reason than in the jump case. It is not that the two sides disagree; it is that neither side approaches a finite value at all.
Asymptote, frozen at c = 0
f(x) = 1/x² climbing off the top of the frame on both sides. No L lines are drawn at all, because neither one-sided limit is finite.
Because the limit is infinite rather than merely two-valued, no patch helps. Assigning f(0) any real number leaves the surrounding values growing arbitrarily large, so condition 3 could never hold. The tool labels this infinite, and marks it explicitly as not removable.
Worth noticing: "L−=L+=+∞" is a statement about how the function behaves, not a claim that the limit exists. In the strict sense the limit fails to exist here — ∞ is a description of the failure, not a value.
Staircase: Discontinuous at Every Integer
The Staircase family is f(x)=⌊x⌋+0.5, the floor function lifted by a half. It is flat on each interval between consecutive integers and steps up by exactly 1 at each integer.
The point of interest the tool jumps to first is c=−2, where L−=−2.5, L+=−1.5, and f(c)=−1.5. That is a jump discontinuity, structurally identical to the jump family — except that here it repeats at every integer, and the jump-to buttons walk through five of them.
Staircase, frozen at c = -2
Each step carries a closed endpoint on the left and an open one on the right. L⁻ = -2.5 and L⁺ = -1.5 straddle the step, and the same picture repeats at every integer.
This family is the useful counterweight to the tidy examples. The others each have one broken point and are continuous everywhere else; this one has infinitely many, one per integer, and is still continuous on every interval strictly between them.
That combination is entirely ordinary. "Continuous" is a statement about a single point, and "continuous on an interval" means continuous at each point of it. A function can fail the test at infinitely many places and still be perfectly well behaved on each piece in between — which is exactly how floor, ceiling, and rounding functions behave in practice.
Probing With the c Slider
DemoProbing with the c slider
Step 0 of 5
The point c slider sweeps the probing position from −3 to +3 in steps of 0.05. As you drag:
• The dashed vertical line on the graph moves to the current c.
• The horizontal dashed lines for L− and L+ relocate to the current one-sided limit values.
• Dots appear on the graph at (c,L−), (c,L+), and (c,f(c)) — so you can see when the three quantities coincide and when they don't.
• Each of the three checklist rows updates with the current numerical value and a pass or fail mark.
For most values of c the checklist passes — every function in the tool is continuous almost everywhere. The interesting points are the isolated values where something fails. The slider is for sweeping past them and watching the checklist flip; the jump-to buttons are for landing on them exactly.
Jump-to Buttons
DemoJump to and Reset
Step 0 of 5
Each family ships with its known points of interest as Jump to buttons beneath the c slider. Clicking one snaps c to that exact value, so you don't have to fight the slider to land precisely on a discontinuity.
• Most families have one or two points of interest — the spot where the limit fails, the spot where f(c) is mis-set, or the asymptote.
• The Staircase family has five — one at each integer in the visible window — so you can walk through a regular sequence of jump discontinuities.
• When c is currently on a point of interest, that button is highlighted; click any other to move.
The Reset button next to the Parameters label returns c to the family's first point of interest (or to zero if the family has none).
Reading the Three-Condition Checklist
The boxed panel below the graph lists the three conditions for continuity at c, each with a green check or a red cross:
• Condition 1: f(c) is defined. Shown with the numerical value when it passes; with the phrase undefined when it fails. Polynomials always pass; rational functions fail wherever the denominator is zero.
• Condition 2: the limit exists. Passes when L−=L+ and both are finite. Fails for jumps (finite but unequal) and infinite discontinuities (at least one limit is not finite).
• Condition 3: f(c) equals the limit. Can only be checked when conditions 1 and 2 both pass. When 1 or 2 has already failed, this row defers to the prior failure.
The final verdict line summarizes the result: continuous when all three pass, otherwise removable, jump, or infinite depending on which condition failed and how.
Display Toggles
DemoDisplay, plot size and tabs
Step 0 of 5
The Display section in the left panel controls which visual elements show on the graph:
• f(x) — toggles the function curves themselves. Off by mistake on first try, but useful when you want to focus on just the limit lines.
• L⁻, L⁺ — toggles the dashed horizontal limit lines and their markers at the point (c,L±). Turn off when probing a continuous region where everything coincides.
• x = c — toggles the dashed vertical line that follows the slider. Off, the visible state is cleaner but you lose the visual anchor for c.
The Accent color picker at the bottom recolors the live highlight throughout — slider track, point marker, callout label, and the verdict border — useful for distinguishing screenshots or for personal preference.
Plot Size, Applied Strip and Info Panel
The S, M, L and XL buttons above the plot set its size, and the readout next to Maximize prints it: S is 500×400, M700×550, L900×850 and XL1100×850. The page opens at 880×460, which is none of the four. The plot never grows wider than its column, so on most screens the buttons mainly change its height. The curve, the limit lines and the markers are redrawn at the new size; the checklist does not change.
The Applied strip at the bottom of the middle column repeats the current value of c. Its last tag, All 3 pass ⇒ continuous, is the rule itself and reads the same at every point; the verdict for the current c is the last line of the three-condition checklist above it.
The info panel on the right has two tabs. Explanation is the live one: the family and the current c, a short description, the equation, the current f(c), L− and L+, which conditions fail, what the slider does, and a closing note that links to the section on that family. Concepts is fixed text on the three-part definition and on continuity over an interval. For the definition itself, see what is continuity.
What Is Continuity
A function f is continuous at the point c when three conditions all hold:
1.f(c) is defined.
2.x→climf(x) exists.
3.x→climf(x)=f(c).
Informally: as x approaches c, the function value approaches f(c). The graph passes through the point (c,f(c)) without a break, hole, or jump.
A function is continuous on an intervalI when it is continuous at every point of I. Polynomials, sinx, cosx, ex, and many other elementary functions are continuous on their entire domain.
For the full theory of continuity and the formal ε–δ definition, see the continuity page.
Types of Discontinuities
When a function fails to be continuous at c, the failure pattern names the discontinuity:
• Removable — the two-sided limit exists, but either f(c) is undefined (condition 1 fails) or f(c) differs from the limit (condition 3 fails). Reassigning the single value f(c) to the limit repairs the function. The Hole and Wrong-value families show this case.
• Jump — both one-sided limits are finite, but unequal. The two-sided limit doesn't exist. No single value of f(c) can repair it. The Jump and Staircase families show this case.
• Infinite — at least one one-sided limit is +∞ or −∞, typically because of a vertical asymptote. Not removable. The Asymptote family shows this case.
For deeper coverage of each kind, see the discontinuity page.
Continuity and Limits
Continuity is built directly on limits. The middle condition — the limit exists — is exactly the condition that limx→c−f(x)=limx→c+f(x) and is finite.
So checking continuity at c amounts to comparing three numbers:
• the left limit L−=limx→c−f(x),
• the right limit L+=limx→c+f(x),
• the value f(c).
If all three are equal and finite, f is continuous at c. Each pattern of agreement and disagreement produces a different kind of discontinuity. The tool displays these three numbers live as you slide c — the same three numbers that the checklist rows compare.
For a tool focused on the limit step alone — including an ε slider and one-sided approach controls — see the limits visualizer.
Related Concepts
Limits — the foundation continuity is built on. One-sided limits and two-sided limits, formal definitions, and limit laws.
Discontinuity — full classification of removable, jump, infinite, and essential discontinuities, with examples of each.
Differentiability — every differentiable function is continuous, but not every continuous function is differentiable. The absolute value function is continuous at 0 but not differentiable there.
Intermediate Value Theorem — continuous functions on closed intervals take every value between their endpoints. The most useful direct consequence of continuity.
Extreme Value Theorem — continuous functions on closed bounded intervals attain a maximum and minimum.
Piecewise functions — the most common source of jump and removable discontinuities. See the piecewise function builder for an interactive view.
Asymptotes — vertical asymptotes are infinite discontinuities; horizontal asymptotes describe end behavior.
Derivative — differentiability is the stronger condition: every differentiable function is continuous, and the explorer shows where that implication stops.