Calculus · Quick Reference

Continuity Cheat Sheet

Differential Calculus · Lesson 2/20

In one line: a function is continuous if you can draw its graph without lifting your pen — no breaks, holes, or jumps.

Key Ideas

1Definition of Continuity. A function f is continuous at x=a if three things hold: f(a) is defined, the limit as x approaches a exists, and that limit equals f(a).

2Types of Discontinuity. Removable (a hole, fixable by redefining one point), jump (left and right limits differ), and infinite (the function shoots to infinity).

3Continuity on an Interval. A function is continuous on an interval if it is continuous at every point within that interval.

4Polynomials and Continuity. All polynomial functions are continuous everywhere. Rational functions are continuous everywhere except where the denominator is zero.

5Intermediate Value Theorem. If f is continuous on [a,b] and N is between f(a) and f(b), there exists some c in [a,b] where f(c) = N.

Worked Examples

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

Yes, continuous at x=2

Is f(x) = 1/(x-3) continuous at x=3?

No, it has an infinite discontinuity at x=3

Use the Intermediate Value Theorem to show f(x)=x^3-x-1 has a root between x=1 and x=2.

A root exists between x=1 and x=2

Practice Yourself

Is f(x)=5x-2 continuous everywhere?

Yes, all polynomials are continuous everywhere

Where is f(x)=1/(x-4) discontinuous?

x = 4

Is f(x)=x^2 continuous at x=0?

Yes

Name the three conditions required for continuity at a point.

f(a) defined, limit exists, limit equals f(a)