Continuous Function Formula

A function is continuous at a point if the limit equals the function value there, with no jumps, holes, or vertical asymptotes in the interval of interest.

The Formula

lim⁡x→af(x)=f(a) for all a in the domain

When to use: A continuous function can be drawn without lifting the pencil — there are no sudden jumps, gaps, or points that shoot to infinity.

Quick Example

f(x)=x2 is continuous. f(x)=1x is not continuous at x=0.

Notation

f is continuous at a means three conditions hold: f(a) is defined, lim⁡x→af(x) exists, and the limit equals f(a).

What This Formula Means

A function is continuous at a point if the limit equals the function value there, with no jumps, holes, or vertical asymptotes in the interval of interest.

A continuous function can be drawn without lifting the pencil — there are no sudden jumps, gaps, or points that shoot to infinity.

Formal View

f is continuous at a   ⟺   ∀ ε>0,  ∃ δ>0:∣x−a∣<δ  ⟹  ∣f(x)−f(a)∣<ε

Worked Examples

Example 1

easy
Show that f(x)=3x2−5x+2 is continuous at x=1 using the three-part definition of continuity.

Answer

f is continuous at x=1

First step

1
Part 1 — f(1) exists: f(1)=3(1)−5(1)+2=0. ✓

Full solution

  1. 2
    Part 2 — lim⁡x→1f(x) exists: since f is a polynomial, the limit equals the function value. lim⁡x→1(3x2−5x+2)=3−5+2=0. ✓
  2. 3
    Part 3 — Limit equals function value: lim⁡x→1f(x)=0=f(1). ✓ All three conditions hold, so f is continuous at x=1.
Continuity requires three conditions: the function value exists, the limit exists, and they are equal. Polynomials satisfy all three at every point, making them everywhere continuous.

Example 2

hard
Find where f(x)=x2−4x−2 is discontinuous, classify the discontinuity, and determine if it can be removed.

Example 3

medium
Apply the IVT to show f(x)=x3+x−4 has a root in [1,2].

Common Mistakes

  • Assuming defined-everywhere means continuous - a function can have a value at every point yet still jump.
  • Overlooking holes from cancelled factors - a removable hole still breaks continuity at that point.
  • Ignoring boundary matching in piecewise functions - the pieces must agree in value at the seams to be continuous.

Why This Formula Matters

Continuity guarantees no surprises — small input changes give small output changes — which is what makes the intermediate value theorem and most of calculus work. A hidden jump or hole breaks guarantees that a model relies on. Recognizing it by "Can the graph be drawn through this point without lifting the pencil?" — rather than by familiar numbers — is what lets a student tell it apart from differentiable function and piecewise function and limit in a mixed problem set.

Frequently Asked Questions

What is the Continuous Function formula?

A function is continuous at a point if the limit equals the function value there, with no jumps, holes, or vertical asymptotes in the interval of interest.

How do you use the Continuous Function formula?

A continuous function can be drawn without lifting the pencil — there are no sudden jumps, gaps, or points that shoot to infinity.

What do the symbols mean in the Continuous Function formula?

f is continuous at a means three conditions hold: f(a) is defined, lim⁡x→af(x) exists, and the limit equals f(a).

Why is the Continuous Function formula important in Math?

Continuity guarantees no surprises — small input changes give small output changes — which is what makes the intermediate value theorem and most of calculus work. A hidden jump or hole breaks guarantees that a model relies on. Recognizing it by "Can the graph be drawn through this point without lifting the pencil?" — rather than by familiar numbers — is what lets a student tell it apart from differentiable function and piecewise function and limit in a mixed problem set.

What do students get wrong about Continuous Function?

The procedure for continuous function is the easy part; the trap is assuming defined-everywhere means continuous. Asking "Can the graph be drawn through this point without lifting the pencil?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Continuous Function formula?

Before studying the Continuous Function formula, you should understand: function definition.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Functions and Graphs: Complete Foundations for Algebra and Calculus →
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