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Defining Continuity
at a Point

Understand the three conditions that make a function continuous at a point, and learn to identify discontinuities with precision.

AP Calculus BC

What Does Continuous Mean?

Informally, a function is continuous at a point if you can draw its graph through that point without lifting your pencil. There are no holes, jumps, or vertical asymptotes at that location.

a f(a) Continuous at x = a No break at the point

Why Continuity Matters

Continuity is the foundation for nearly every major theorem in calculus. Without it, key results break down.

Intermediate Value Theorem
Guarantees a continuous function hits every value between \( f(a) \) and \( f(b) \).
Extreme Value Theorem
A continuous function on \( [a,b] \) must attain an absolute max and min.
Fundamental Theorem of Calculus
Requires continuity on the integration interval to connect derivatives and integrals.

Quick Review: Limits

Continuity is built on the concept of a limit. Recall that \(\displaystyle\lim_{x \to a} f(x) = L\) means the output values of \( f \) get arbitrarily close to \( L \) as \( x \) approaches \( a \) from both sides.

Left-Hand Limit
\(\displaystyle\lim_{x \to a^-} f(x)\): approach \( a \) from values less than \( a \).
Right-Hand Limit
\(\displaystyle\lim_{x \to a^+} f(x)\): approach \( a \) from values greater than \( a \).

The Three Conditions for Continuity

A function \( f \) is continuous at \( x = a \) if and only if all three of these conditions hold:

1
\( f(a) \) is defined. The point \( a \) must be in the domain of \( f \).
2
\(\displaystyle\lim_{x \to a} f(x) \) exists. Both one-sided limits must agree on a single finite value.
3
\(\displaystyle\lim_{x \to a} f(x) = f(a) \). The limit equals the actual function value at \( a \).

Visualizing the Definition

Continuous a Removable a Jump a
✓ Continuous
All three conditions met: no break at the point.
Removable
Limit exists but \( f(a) \) is missing or misplaced (condition 3 fails).
Jump
One-sided limits differ, so the two-sided limit doesn't exist (condition 2 fails).

Types of Discontinuity

Removable
A "hole" in the graph. The limit exists, but either \( f(a) \) is undefined or \( f(a) \neq \displaystyle\lim_{x \to a} f(x) \). Can be "fixed" by redefining \( f(a) \).
Jump
Left-hand and right-hand limits both exist but are not equal: \(\displaystyle\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)\). Common in piecewise functions.
Infinite
At least one one-sided limit is \( \pm\infty \). The graph has a vertical asymptote at \( x = a \). Example: \( \frac{1}{x} \) at \( x = 0 \).

Continuity vs. Differentiability

These two concepts are closely related but not the same. Every differentiable function is continuous, but a continuous function is not necessarily differentiable.

Continuity at \( a \) Differentiability at \( a \)
Requires \(\displaystyle\lim_{x \to a} f(x) = f(a)\) \(\displaystyle\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\) exists
Graph looks like No break or hole at \( a \) Smooth, no corner or cusp at \( a \)
Implication Necessary for differentiability Guarantees continuity
Classic counterexample \( |x| \) is continuous at \( 0 \) \( |x| \) is NOT differentiable at \( 0 \)

How to Check Continuity

Follow this systematic checklist every time you need to determine whether \( f \) is continuous at \( x = a \).

1
Evaluate \( f(a) \). Is the function defined at the point? If \( a \) is not in the domain, stop: \( f \) is discontinuous.
2
Compute one-sided limits. Find \(\displaystyle\lim_{x \to a^-} f(x)\) and \(\displaystyle\lim_{x \to a^+} f(x)\). If they differ, the two-sided limit does not exist.
3
Check the two-sided limit. If both one-sided limits equal the same value \( L \), then \(\displaystyle\lim_{x \to a} f(x) = L\).
4
Compare. Verify \( f(a) = L \). If yes, \( f \) is continuous at \( a \). If not, it has a removable discontinuity.

Example 1: Polynomial

Is \( f(x) = x^2 - 3x + 2 \) continuous at \( x = 1 \)?

1
\( f(1) = 1 - 3 + 2 = 0 \). The function is defined.
2
\(\displaystyle\lim_{x \to 1} (x^2 - 3x + 2) = 0 \). Polynomials have limits by direct substitution.
3
\(\displaystyle\lim_{x \to 1} f(x) = f(1) = 0 \). All three conditions satisfied.

Example 2: Removable Discontinuity

Is \(\displaystyle g(x) = \frac{x^2 - 4}{x - 2}\) continuous at \( x = 2 \)?

1
\( g(2) \) is undefined because \( x - 2 = 0 \) creates division by zero. Condition 1 fails immediately.
2
Factor: \(\displaystyle g(x) = \frac{(x-2)(x+2)}{x-2} = x + 2\) for \( x \neq 2 \). So \(\displaystyle\lim_{x \to 2} g(x) = 4\). The limit exists.
3
We cannot compare \( g(2) \) with the limit because \( g(2) \) does not exist.

Graphing the Hole

x y 1 2 3 2 3 4 5 Hole at (2, 4) g(2) undefined

Example 3: Piecewise Function

Is \( h(x) \) continuous at \( x = 1 \)?

1
\( h(1) = 5 \). The function is defined. ✓
2
Left: \(\displaystyle\lim_{x \to 1^-} (2x+1) = 3\). Right: \(\displaystyle\lim_{x \to 1^+} (x^2+2) = 3\). Both equal 3. ✓
3
Compare: \(\displaystyle\lim_{x \to 1} h(x) = 3 \neq 5 = h(1)\). Condition 3 fails. ✕

Graphing the Piecewise Function

x y 1 2 1 3 5 (1, 5) Limit = 3

Common Mistakes

✕ Skipping Condition 1
Students find the limit but forget to check that \( f(a) \) is actually defined. If the point is not in the domain, the function is automatically discontinuous.
✕ Only Checking One Side
Computing only \(\displaystyle\lim_{x \to a^+} f(x)\) or only \(\displaystyle\lim_{x \to a^-} f(x)\) and declaring the limit exists. Both sides must agree.
✕ Confusing "Limit Exists" with "Continuous"
Even if the limit exists, \( f \) can still be discontinuous if \( f(a) \neq \displaystyle\lim_{x \to a} f(x)\). All three conditions must hold.

Summary of Discontinuity Types

Type \( f(a) \) Defined? Limit Exists? Limit = \( f(a) \)?
Continuous ✓ Yes ✓ Yes ✓ Yes
Removable Maybe ✓ Yes ✕ No
Jump Maybe ✕ No ✕ N/A
Infinite ✕ No ✕ No ✕ N/A

Question 1 of 5

Which of the following is not a required condition for \( f \) to be continuous at \( x = a \)?

Question 2 of 5

The function \(\displaystyle g(x) = \frac{x^2 - 9}{x - 3}\) has what type of discontinuity at \( x = 3 \)?

Question 3 of 5

For the piecewise function below, what value of \( k \) makes \( f \) continuous at \( x = 2 \)? $$f(x) = \begin{cases} 3x - 1, & x \le 2 \\ kx^2, & x > 2 \end{cases}$$

Question 4 of 5

If \(\displaystyle\lim_{x \to 4} f(x) = 7\) and \( f(4) = 7 \), which statement is true?

Question 5 of 5

Consider \(\displaystyle f(x) = \begin{cases} x + 3, & x < 1 \\ 2, & x = 1 \\ 5 - x, & x > 1 \end{cases}\). Which condition of continuity fails at \( x = 1 \)?