Defining Continuity at a Point

Introduction

students, imagine watching a live video stream on your phone 📱. If the signal stays steady, the video looks smooth. If there is a glitch, the picture freezes or jumps. In calculus, continuity describes that same idea for a function: does the graph flow smoothly through a point, or does it break? This lesson focuses on defining continuity at a point, a key idea in AP Calculus BC under Limits and Continuity.

By the end of this lesson, you should be able to:

Explain what it means for a function to be continuous at a point.
Use the definition of continuity to check whether a function is continuous.
Connect continuity to limits, function values, and one-sided behavior.
Recognize how continuity fits into bigger ideas like the Intermediate Value Theorem and limits at infinity.

Continuity is not just about pretty graphs. It helps determine whether a model makes sense in real life, such as temperature changes, motion, or population growth. When a function is continuous, its value at a point matches what nearby values suggest should happen.

What It Means for a Function to Be Continuous

A function $f$ is continuous at a point $x=c$ if three things all happen:

$f(c)$ is defined.
$\lim_{x\to c} f(x)$ exists.
$\lim_{x\to c} f(x)=f(c)$.

These three conditions work together. If even one is missing, the function is not continuous at $x=c$.

Think of it like crossing a bridge 🚗. The bridge must exist at that spot, the road must lead smoothly to that spot from both sides, and the surface at the spot must match the incoming road. If the bridge is missing, broken, or mismatched, the trip is not continuous.

Here is the formal definition in symbols:

$$f \text{ is continuous at } x=c \text{ if } \lim_{x\to c} f(x)=f(c).$$

This compact statement includes the three conditions above, because for the equality to make sense, $f(c)$ must exist and the limit must exist first.

Why Each Condition Matters

If $f(c)$ is not defined, there is no actual output at that input.
If $\lim_{x\to c} f(x)$ does not exist, then nearby values do not settle toward one number.
If $\lim_{x\to c} f(x)\neq f(c)$, then the function has a mismatch between the value at the point and the nearby behavior.

For example, suppose a function has a hole at $x=2$. Even if the graph approaches the same $y$-value from both sides, the function is not continuous at $x=2$ unless the point is actually filled in with that same value.

Checking Continuity with Examples

Let’s test a simple function:

$$f(x)=x^2$$

Is $f$ continuous at $x=3$? Yes.

$f(3)=3^2=9$.
$\lim_{x\to 3} x^2=9$.
Since $\lim_{x\to 3} f(x)=f(3)$, the function is continuous at $x=3$.

Polynomials are continuous everywhere, so $x^2$, $x^3-4x+1$, and similar expressions are continuous for all real $x$.

Now try a piecewise function:

$$g(x)=\begin{cases}

2x+1, & x<1 \\

$5, & x=1 \\$

4x-1, & x>1

$\end{cases}$$$

Check continuity at $x=1$:

$g(1)=5$.
Left-hand limit: $\lim_{x\to 1^-} g(x)=\lim_{x\to 1^-}(2x+1)=3$.
Right-hand limit: $\lim_{x\to 1^+} g(x)=\lim_{x\to 1^+}(4x-1)=3$.
So $\lim_{x\to 1} g(x)=3$.

Because $\lim_{x\to 1} g(x)=3\neq 5=g(1)$, the function is not continuous at $x=1$. The graph has a mismatch: the nearby values head toward $3$, but the actual function value jumps to $5$.

This kind of example is common on AP Calculus BC because you may need to identify the exact reason a function fails to be continuous.

Continuity and Limits Work Together

Continuity is really a special kind of limit statement. A limit tells you what a function is approaching near a point, while continuity tells you that the function actually reaches that same value at the point.

That means continuity is stronger than just having a limit.

For instance, consider a removable discontinuity:

$$h(x)=\frac{x^2-4}{x-2}$$

for $x\neq 2$.

Factor the numerator:

$$h(x)=\frac{(x-2)(x+2)}{x-2}=x+2, \quad x\neq 2$$

Then:

$$\lim_{x\to 2} h(x)=4$$

But $h(2)$ is undefined because the original formula gives division by $0$. So the function is not continuous at $x=2$.

However, if we define a new function

$$k(x)=\begin{cases}

$\frac{x^2-4}{x-2}, & x\neq 2 \\$

$4, & x=2$

$\end{cases}$$$

then $k$ is continuous at $x=2$ because now:

$k(2)=4$
$\lim_{x\to 2} k(x)=4$
so $\lim_{x\to 2} k(x)=k(2)$

This process is called removing a discontinuity by redefining the function.

Types of Discontinuities You Should Recognize

students, AP Calculus BC expects you to recognize common discontinuity types:

1. Removable Discontinuity

A hole in the graph. The limit exists, but the function value is missing or different.

Example:

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

for $x\neq 1$.

The graph behaves like $x+1$, but there is a hole at $x=1$.

2. Jump Discontinuity

The left-hand and right-hand limits both exist, but they are not equal.

Example:

A piecewise function might approach $2$ from the left and $5$ from the right at the same $x$-value.

3. Infinite Discontinuity

The function grows without bound near a point, often creating a vertical asymptote.

Example:

$$f(x)=\frac{1}{x-3}$$

At $x=3$, the function is not continuous because values blow up toward $\infty$ or $-\infty$.

These patterns matter because a continuous function cannot have holes, jumps, or vertical blow-ups at the point in question.

Continuity on an Interval and Real-World Meaning

A function is continuous on an interval if it is continuous at every point in that interval. For example, $f(x)=x^2$ is continuous on $(-\infty,\infty)$, while $f(x)=\frac{1}{x}$ is continuous on $(-\infty,0)$ and $(0,\infty)$ but not at $x=0$.

In real life, continuity means no sudden impossible breaks in the model. For example:

A car’s position as it moves is usually continuous because it cannot teleport 🚗.
Temperature over time is often modeled as continuous because it changes gradually.
A staircase model, however, may be discontinuous because the height changes in jumps.

AP Calculus uses continuity to justify many powerful results. One of the most important is the Intermediate Value Theorem, which says that if a function is continuous on an interval and takes two different values, then it must take every value between them somewhere in that interval. This works only because continuity prevents gaps and jumps.

Quick Continuity Checklist

When you check continuity at $x=c$, ask:

Is $f(c)$ defined?
Does $\lim_{x\to c} f(x)$ exist?
Is $\lim_{x\to c} f(x)=f(c)$?

If the answer to all three is yes, then the function is continuous at $x=c$.

For piecewise functions, always check both sides. For rational functions, look for zeros in the denominator. For radicals, make sure the expression stays defined. For trig functions, remember that many are continuous where their formulas make sense.

Conclusion

Continuity at a point means a function behaves smoothly there with no gaps, jumps, or mismatches. The key idea is simple but powerful: the limit near the point must exist and must equal the actual function value at that point. This idea connects limits, graph behavior, and real-world modeling.

students, mastering continuity helps you solve AP Calculus BC problems about function behavior, piecewise definitions, and theorems like the Intermediate Value Theorem. It also prepares you for deeper topics where continuity is required for more advanced calculus results.

Study Notes

A function $f$ is continuous at $x=c$ if $f(c)$ is defined, $\lim_{x\to c} f(x)$ exists, and $\lim_{x\to c} f(x)=f(c)$.
Continuity means the graph has no break at the point.
A removable discontinuity is a hole where the limit exists but the function value is missing or different.
A jump discontinuity happens when the left-hand and right-hand limits are different.
An infinite discontinuity happens when the function grows without bound near the point.
Polynomials are continuous everywhere.
Rational functions are continuous wherever their denominators are not zero.
Piecewise functions must be checked carefully at the boundary points.
Continuity is essential for the Intermediate Value Theorem.
In modeling, continuity often represents smooth, realistic change over time 📈.