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Defining Continuity
at a Point

Master the three-part definition of continuity that forms the foundation of calculus: limits, function values, and their perfect agreement.

AP Calculus AB

What Does Continuous Mean?

Intuitively, 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.

Continuous vs. Discontinuous

Continuous a Discontinuous a

Why Continuity Matters

Foundation of Calculus
The theorems you rely on: the Intermediate Value Theorem, Extreme Value Theorem, and differentiability all require continuity.
Direct Substitution
When \(f\) is continuous at \(a\), evaluating \(\lim_{x \to a} f(x)\) is as simple as computing \(f(a)\).
Real-World Models
Physical quantities like temperature, position, and pressure are modeled by continuous functions: no instantaneous teleportation.

The Three Conditions for Continuity

1
\(f(a)\) is defined. The function must have an actual output value at the point \(x = a\).
2
\(\lim_{x \to a} f(x)\) exists. The left-hand and right-hand limits must agree, producing a single finite value.
3
\(\lim_{x \to a} f(x) = f(a)\). The limit value and the function value must be equal.

The Formal Definition

This compact statement is what the AP exam expects you to cite. Every continuity question reduces to checking whether this equation holds.

Visualizing Each Condition

Condition 1 Fails
\(f(a)\) undefined
 a Condition 2 Fails
\(\lim_{x \to a} f(x)\) DNE (jump)
 a Condition 3 Fails
\(\lim_{x \to a} f(x) \neq f(a)\)
 a

Each panel shows a different condition failing. Left: a hole with no value. Center: a jump where left and right limits disagree. Right: the value exists but doesn't match the limit.

Types of Discontinuity

Removable
The limit exists but \(f(a)\) is either undefined or different from the limit. A "hole" you can fill by redefining \(f(a)\).
Jump
Left-hand and right-hand limits both exist but are not equal. Common in piecewise functions.
Infinite
At least one side approaches \(\pm\infty\). The function has a vertical asymptote at \(x = a\).

Removable vs. Non-Removable

Removable Non-Removable
Limit exists? Yes No (or one-sided only)
Graph appearance Hole (open circle) Jump or vertical asymptote
Can be "fixed"? Yes: redefine \(f(a)\) No: fundamental break
Example \(\frac{x^2 - 1}{x - 1}\) at \(x = 1\) \(\frac{1}{x}\) at \(x = 0\)

How to Test Continuity at \(x = a\)

1
Evaluate \(f(a)\). Substitute \(x = a\) into the function. If the result is undefined (division by zero, etc.), stop: the function is discontinuous.
2
Find \(\lim_{x \to a} f(x)\). Compute both the left-hand limit \(\lim_{x \to a^-} f(x)\) and right-hand limit \(\lim_{x \to a^+} f(x)\). If they differ, the limit does not exist.
3
Compare. Check whether \(\lim_{x \to a} f(x) = f(a)\). If equal, the function is continuous at \(a\). If not, it is discontinuous.

Example 1: Polynomial

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

1
\(f(1) = (1)^2 - 3(1) + 2 = 0\). The value is defined. ✓
2
\(\lim_{x \to 1} (x^2 - 3x + 2) = 0\). Polynomials have limits via direct substitution. ✓
3
\(\lim_{x \to 1} f(x) = 0 = f(1)\). They match. ✓

Example 2: Rational Function

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

1
\(g(2) = \frac{0}{0}\): undefined. ✕ Condition 1 fails.
2
Factor: \(\frac{(x-2)(x+2)}{x-2} = x+2\) for \(x \neq 2\). So \(\lim_{x \to 2} g(x) = 4\). The limit exists.
3
Cannot compare: \(g(2)\) is undefined, so condition 3 automatically fails too.

Example 3: Piecewise Function

Is the following function continuous at \(x = 3\)?

1
\(h(3) = 3^2 - 4 = 5\). Defined. ✓
2
Left: \(\lim_{x \to 3^-}(2x-1) = 5\). Right: \(\lim_{x \to 3^+}(x^2-4) = 5\). Both equal 5. ✓
3
\(\lim_{x \to 3} h(x) = 5 = h(3)\). ✓ Continuous at \(x = 3\).

Example 4: Jump Discontinuity

1
\(p(2) = 2(2) - 3 = 1\). Defined. ✓
2
Left: \(\lim_{x \to 2^-}(x+1) = 3\). Right: \(\lim_{x \to 2^+}(2x-3) = 1\). Since \(3 \neq 1\), the limit does not exist.
3
✕ Condition 2 fails. This is a jump discontinuity and is non-removable.

Continuous Everywhere: Key Families

Polynomials
Continuous on \((-\infty, \infty)\). Direct substitution always works.
Trig Functions
\(\sin x\) and \(\cos x\) are continuous on all reals. \(\tan x\) is continuous on its domain.
Exponentials
\(e^x\), \(a^x\) are continuous on \((-\infty, \infty)\). \(\ln x\) is continuous on \((0, \infty)\).

Common Mistakes

✕ Forgetting All 3 Conditions
Students correctly find that the limit exists, but forget to check if \(f(a)\) is defined and if it actually equals the limit.
✕ Ignoring One-Sided Limits
For piecewise functions, you must check both \(\lim_{x \to a^-} f(x)\) and \(\lim_{x \to a^+} f(x)\) separately.
✕ Confusing "Defined" with "Continuous"
A function can have \(f(a) = 7\) yet be discontinuous if the limit is different or doesn't exist.

Question 1 of 5

Which of the following is NOT one of the three conditions for continuity at \(x = a\)?

Question 2 of 5

Let \(f(x) = \dfrac{x^2 - 9}{x - 3}\). At \(x = 3\), the function has a:

Question 3 of 5

Given the piecewise function:
\(g(x) = \begin{cases} 3x + 1 & \text{if } x < 2 \\ k & \text{if } x = 2 \\ x^2 + 3 & \text{if } x > 2 \end{cases}\)
What value of \(k\) makes \(g\) continuous at \(x = 2\)?

Question 4 of 5

If \(\lim_{x \to 5} f(x) = 3\) and \(f(5) = 3\), which statement is true?

Question 5 of 5

At which point is \(f(x) = \dfrac{1}{x - 4}\) discontinuous?