Continuous Function Formula

The Formula

\lim_{x \to a} f(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) = x^2 is continuous. f(x) = \frac{1}{x} is not continuous at x = 0.

Notation

f is continuous at a means three conditions hold: f(a) is defined, \lim_{x \to a} f(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 \iff \forall\,\varepsilon > 0,\;\exists\,\delta > 0: |x - a| < \delta \implies |f(x) - f(a)| < \varepsilon

Worked Examples

Example 1

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

Solution

  1. 1
    Part 1 — f(1) exists: f(1) = 3(1)-5(1)+2 = 0. ✓
  2. 2
    Part 2 — \lim_{x\to1} f(x) exists: since f is a polynomial, the limit equals the function value. \lim_{x\to1}(3x^2-5x+2) = 3-5+2 = 0. ✓
  3. 3
    Part 3 — Limit equals function value: \lim_{x\to1}f(x) = 0 = f(1). ✓ All three conditions hold, so f is continuous at x=1.

Answer

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) = \dfrac{x^2 - 4}{x - 2} is discontinuous, classify the discontinuity, and determine if it can be removed.

Common Mistakes

  • Thinking continuous means smooth — |x| is continuous everywhere but has a sharp corner at x = 0
  • Assuming all functions defined by formulas are continuous — f(x) = \frac{1}{x} is not continuous at x = 0
  • Believing discontinuities are always visible on a graph — removable discontinuities (holes) may be invisible at normal zoom levels

Why This Formula Matters

Continuous functions are predictable—small input changes mean small output changes.

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 \to a} f(x) exists, and the limit equals f(a).

Why is the Continuous Function formula important in Math?

Continuous functions are predictable—small input changes mean small output changes.

What do students get wrong about Continuous Function?

A function can be continuous everywhere except certain points.

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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