Continuous Function

Q: How do you tell if a function is continuous from its graph?

Trace the graph with your finger from left to right. If you never have to lift your finger — no gaps, jumps, or holes — the function is continuous over that interval. A single hole or jump at any point means the function is discontinuous there.

Q: What are the three conditions for continuity at a point?

For a function f to be continuous at x = c, three things must all be true: (1) f(c) must be defined, (2) the limit of f(x) as x approaches c must exist, and (3) that limit must equal f(c). If any one condition fails, f is discontinuous at c.

Continuous Function

A function with a connected graph.

Two graphs: left shows a continuous U-shaped curve on x-y axes; right shows a discontinuous curve with an open and closed point.

Definition of Continuity: f(x) is continuous at x=a if: 1. lim x→a f(x) exists, 2. f(a) exists, 3. lim x→a f(x)=f(a)

See also

Discontinuous function

Key Formula

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

Where:

$f$ = The function being tested for continuity
$c$ = The specific point where continuity is being checked
$\lim_{x \to c} f(x)$ = The limit of f(x) as x approaches c, which must exist and equal f(c)

Worked Example

Problem: Determine whether the function f(x) = x² + 1 is continuous at x = 2.

Step 1: Check that f(c) is defined. Evaluate f(2).

f(2) = 2^2 + 1 = 5

Step 2: Check that the limit of f(x) as x approaches 2 exists. Substitute values approaching 2 from both sides, or evaluate directly for a polynomial.

\lim_{x \to 2} (x^2 + 1) = 4 + 1 = 5

Step 3: Check that the limit equals the function value.

\lim_{x \to 2} f(x) = 5 = f(2)

Answer: All three conditions are satisfied, so f(x) = x² + 1 is continuous at x = 2. In fact, every polynomial is continuous at every real number.

Another Example

Problem: Determine whether the piecewise function g(x) = { x + 3 if x < 1, and 2x if x ≥ 1 } is continuous at x = 1.

Step 1: Evaluate g(1). Since 1 ≥ 1, use the second piece.

g(1) = 2(1) = 2

Step 2: Find the left-hand limit as x approaches 1 from below, using the first piece.

\lim_{x \to 1^-} (x + 3) = 1 + 3 = 4

Step 3: Find the right-hand limit as x approaches 1 from above, using the second piece.

\lim_{x \to 1^+} 2x = 2(1) = 2

Step 4: Compare the left- and right-hand limits. Since 4 ≠ 2, the two-sided limit does not exist.

\lim_{x \to 1^-} g(x) = 4 \neq 2 = \lim_{x \to 1^+} g(x)

Answer: The function g(x) is NOT continuous at x = 1 because the left-hand and right-hand limits are different. There is a jump discontinuity at that point.

Frequently Asked Questions

How do you tell if a function is continuous from its graph?

Trace the graph with your finger from left to right. If you never have to lift your finger — no gaps, jumps, or holes — the function is continuous over that interval. A single hole or jump at any point means the function is discontinuous there.

What are the three conditions for continuity at a point?

For a function f to be continuous at x = c, three things must all be true: (1) f(c) must be defined, (2) the limit of f(x) as x approaches c must exist, and (3) that limit must equal f(c). If any one condition fails, f is discontinuous at c.

Continuous Function vs. Discontinuous Function

A continuous function has no breaks, holes, or jumps at any point in its domain — you can draw its graph in one unbroken stroke. A discontinuous function fails at least one of the three continuity conditions at one or more points. Common types of discontinuity include jump discontinuities (the function leaps to a different value), removable discontinuities (a single hole in the graph), and infinite discontinuities (the function shoots toward infinity, as with a vertical asymptote).

Why It Matters

Continuity is a foundational requirement for most of calculus. The Intermediate Value Theorem, for instance, guarantees that a continuous function takes on every value between f(a) and f(b), which is how we prove certain equations have solutions. Many real-world models — temperature over time, the position of a moving object — are naturally continuous, so understanding continuity helps you decide which mathematical tools apply.

Common Mistakes

Mistake: Assuming a function is continuous just because it has a formula or equation.

Correction: Rational functions like f(x) = 1/x have formulas but are not continuous everywhere — this one is undefined (and therefore discontinuous) at x = 0. Always check the domain for points where the function might break.

Mistake: Checking only the function value and ignoring the limit, or vice versa.

Correction: Continuity requires both: the limit as x → c must exist AND it must equal f(c). A function can be defined at a point yet still be discontinuous there if the limit disagrees with the function value (a removable discontinuity).

Related Terms

Function — A continuous function is a special type of function
Discontinuous Function — A function that fails the continuity test
Graph of an Equation or Inequality — Visual tool for checking continuity
Limit — Core concept used to define continuity
Intermediate Value Theorem — Key theorem requiring continuity
Piecewise Function — Often tested for continuity at boundaries
Polynomial — All polynomials are continuous everywhere