Continuity — Definition, Types & Examples

A function is continuous at $x=a$ when the value at $a$ matches the value the function approaches near $a$. In calculus terms, continuity at a point means $f(a)$ exists, $\lim_{x \to a} f(x)$ exists, and those two values are equal.

Written as conditions:

$$f(a) \text{ is defined}, \qquad \lim_{x \to a} f(x) \text{ exists}, \qquad \lim_{x \to a} f(x) = f(a).$$

If even one condition fails, the function is not continuous at that point.

Continuity Definition In Plain English

You may hear continuity described as "drawing the graph without lifting your pencil." That picture helps, but the real definition is about nearby inputs and outputs.

If $x$ moves closer to $a$, then $f(x)$ should move closer to the actual output $f(a)$. That is why continuity depends on both the limit and the function value. A graph can look almost connected and still fail the definition if there is a hole or jump at the point.

How To Check Continuity At A Point

Most problems reduce to the same checklist.

1. Make sure $f(a)$ is defined.
2. Find $\lim_{x \to a} f(x)$.
3. If the left-hand and right-hand limits are different, stop: the function is not continuous there.
4. If the limit exists, compare it with $f(a)$.

This is the practical form of the definition. For polynomials, the check is usually immediate because they are continuous for every real $x$. For rational functions, the likely trouble spots are values that make the denominator zero.

Continuity At A Point, On An Interval, And From One Side

In many classes, "types of continuity" means the setting where you test it.

Continuity at a point means the definition holds at one specific value, such as $x=2$.

Continuity on an interval means the function is continuous at every point in that interval. On a closed interval $[a,b]$, the endpoints are checked with one-sided limits.

One-sided continuity matters at endpoints or piecewise boundaries. For example, continuity from the right at $a$ uses $\lim_{x \to a^+} f(x)$.

You will also see "types" used for the common ways continuity fails: removable, jump, and infinite discontinuities.

Types Of Discontinuity

A removable discontinuity happens when the limit exists but the function value is missing or does not match it. This is the classic hole in the graph.

A jump discontinuity happens when the left-hand and right-hand limits both exist but are different.

An infinite discontinuity happens when the function grows without bound near the point, so there is no finite limit there.

These distinctions matter because not every break behaves the same way. A hole can sometimes be fixed by redefining one value. A jump or vertical asymptote cannot be repaired that way.

Worked Example: Is This Function Continuous At $x=1$?

Consider

$$f(x)= \begin{cases} \frac{x^2-1}{x-1}, & x \ne 1 \\ 2, & x=1 \end{cases}$$

We want to test continuity at $x=1$.

First check the function value. Since the second line defines the point,

$$f(1)=2.$$

Now find the limit. For $x \ne 1$,

$$\frac{x^2-1}{x-1}=\frac{(x-1)(x+1)}{x-1}=x+1.$$

So near $x=1$, the function behaves like $x+1$, which gives

$$\lim_{x \to 1} f(x)=\lim_{x \to 1} (x+1)=2.$$

The limit exists, and it matches the function value:

$$\lim_{x \to 1} f(x)=f(1)=2.$$

So the function is continuous at $x=1$.

This example shows the key condition clearly: fixing a hole only works if you fill it with the same value the limit approaches. Here, the piecewise definition sets $f(1)=2$, which matches the limit, so the function is continuous at $x=1$.

Common Mistakes When Testing Continuity

1. Checking only whether $f(a)$ exists. A defined value alone does not guarantee continuity.
2. Checking only the limit. The limit can exist even when the function value is different or missing.
3. Forgetting one-sided limits for piecewise functions. If the two sides disagree, the function is not continuous there.
4. Assuming every familiar-looking formula is continuous everywhere. Rational functions can fail where the denominator is zero.

When Continuity Is Used In Calculus

Continuity matters because many major calculus results assume it. The Intermediate Value Theorem, for example, requires continuity on an interval. Differentiability is even stronger: if a function is differentiable at a point, then it must be continuous there.

Outside theorem statements, continuity helps you decide whether substitution is valid, whether a graph has a real break, and whether a model changes gradually or suddenly.

Try A Similar Problem

Try your own version with a piecewise function at the boundary point. Compute the left-hand limit, the right-hand limit, and the actual function value separately. If you want the next step, explore limits and notice that continuity is really the moment when the limit and the function value agree.