Continuity — Definition, Types & Examples

A function is continuous at $x=a$ when the value at $a$ matches the value the function approaches near $a$. You test continuity whenever you need to know whether substitution is valid, whether a graph has a real break, or whether a calculus theorem that requires continuity can be applied.

Written as conditions, continuity at a point means:

If even one condition fails, the function is not continuous at that point. You may hear continuity described as "drawing the graph without lifting your pencil," but the real definition is about nearby inputs and outputs: as $x$ moves closer to $a$, $f(x)$ should move closer to the actual output $f(a)$. A graph can look almost connected and still fail if there is a hole or jump at the point.

When You Test Continuity, And In Which Setting

Continuity comes up at a point, on an interval, or from one side.

Continuity at a point means the three conditions hold 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:

• A removable discontinuity: the limit exists but the function value is missing or does not match it. This is the classic hole in the graph, sometimes fixable by redefining one value.
• A jump discontinuity: the left-hand and right-hand limits both exist but are different.
• An infinite discontinuity: the function grows without bound near the point, so there is no finite limit there.

A hole can sometimes be repaired by redefining a value. A jump or vertical asymptote cannot.

The Steps To Check Continuity At A Point

1. Check the function value

Make sure $f(a)$ exists at the point you are testing.

2. Find the limit

Evaluate $\lim_{x \to a} f(x)$. If the left-hand and right-hand limits are different, stop: the function is not continuous there.

3. Match limit and value

The function is continuous at $a$ only if the limit exists and equals $f(a)$.

4. Name the break if needed

If the conditions fail, decide whether the issue is removable, jump, or infinite.

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.

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

Consider

First check the function value. The second line defines the point, so

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

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

The limit exists, and it matches the function value:

So the function is continuous at $x=1$. This shows the key condition: 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.

Where Each Step Goes Wrong

Each condition has a matching trap:

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.

To rehearse the full routine, take a piecewise function at its boundary point and compute the left-hand limit, the right-hand limit, and the actual function value separately. Continuity is really the moment when the limit and the function value agree.

Why It Matters

Continuity underlies many major calculus results. The Intermediate Value Theorem requires continuity on an interval, and differentiability is even stronger: if a function is differentiable at a point, then it must be continuous there. Outside theorem statements, continuity tells you whether substitution is valid and whether a model changes gradually or suddenly.