Continuity at a Point

Continuity at a point means that a function has no breaks, jumps, or holes at that specific

x

-value. For a function to be continuous at a point, it must be defined there, its limit must exist there, and the limit must equal the function's value.

A function $f$ is continuous at a point $x = c$ if three conditions are all satisfied: (1) $f(c)$ is defined, (2) $\lim_{x \to c} f(x)$ exists, and (3) $\lim_{x \to c} f(x) = f(c)$ . If any one of these conditions fails, the function is discontinuous at $x = c$ . This definition captures the intuitive idea that you can draw the graph through the point without lifting your pen.

Key Formula

\lim_{x \to c} f(x) = f(c)

Where:

$f$ = the function being tested for continuity
$c$ = the specific x-value (point) where continuity is being checked
$f(c)$ = the value of the function at x = c

Worked Example

Problem: Determine whether

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

is continuous at

x = 2

Step 1 — Check if f(2) is defined: Substitute

x = 2

into the function.

f(2) = \frac{2^2 - 4}{2 - 2} = \frac{0}{0}

Step 2 — Conclude from Condition 1:

f(2)

is undefined because we get division by zero. Since the first condition already fails, the function is not continuous at

x = 2

Step 3 — Investigate the limit anyway: Factor the numerator to see what the limit would be, which helps you understand the type of discontinuity.

\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2}(x+2) = 4

Step 4 — Identify the discontinuity type: The limit exists and equals 4, but

f(2)

is undefined. This means there is a removable discontinuity (a hole) at

x = 2

. If we redefined

f(2) = 4

, the function would become continuous there.

Answer:

f(x)

is not continuous at

x = 2

because

f(2)

is undefined, even though the limit exists.

Why It Matters

Continuity at a point is foundational to calculus. Many key theorems — including the Intermediate Value Theorem and the rules for differentiation — require functions to be continuous. When you check differentiability at a point or evaluate a definite integral, continuity is often the first thing you need to verify.

Common Mistakes

Mistake: Checking only whether the limit exists and forgetting to verify that the function is actually defined at the point.

Correction: All three conditions must hold. A function can have a limit at

x = c

but still be discontinuous there if

f(c)

is undefined or doesn't match the limit.

Mistake: Assuming a function is continuous just because you can compute

f(c)

and get a number.

Correction: You also need the limit to exist and equal

f(c)

. Piecewise functions, for example, can be defined at a point yet have a jump where the left- and right-hand limits disagree.

Related Terms

Continuous Function — Continuous everywhere vs. at a single point
Discontinuity — What occurs when continuity conditions fail
Limit — Central requirement in the continuity definition