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 limit f(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.
Infinite discontinuity — a point where at least one one-sided limit is +∞ or −∞. Typically a vertical asymptote.
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
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.
Probing With the c Slider
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
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
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.
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.