01 Key Concepts
Definition 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).
Types of Discontinuity
Removable (a hole, fixable by redefining one point), jump (left and right limits differ), and infinite (the function shoots to infinity).
Continuity on an Interval
A function is continuous on an interval if it is continuous at every point within that interval.
Polynomials and Continuity
All polynomial functions are continuous everywhere. Rational functions are continuous everywhere except where the denominator is zero.
Intermediate 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.
02 Solved Examples
Example 1 Is f(x) = x^2 + 3 continuous at x=2?
- f(2) = 4+3 = 7, which is defined.
- lim as x approaches 2 of (x^2+3) = 7, using direct substitution since it's a polynomial.
- The limit equals f(2), so continuity holds.
Answer: Yes, continuous at x=2
Example 2 Is f(x) = 1/(x-3) continuous at x=3?
- f(3) = 1/0, which is undefined.
- Since f(3) is not defined, the first condition for continuity already fails.
Answer: No, it has an infinite discontinuity at x=3
Example 3 Use the Intermediate Value Theorem to show f(x)=x^3-x-1 has a root between x=1 and x=2.
- f(1) = 1-1-1 = -1 (negative).
- f(2) = 8-2-1 = 5 (positive).
- Since f is continuous (a polynomial) and changes sign, IVT guarantees a root between 1 and 2.
Answer: A root exists between x=1 and x=2
03 Practice Questions
1Is f(x)=5x-2 continuous everywhere?
Yes, all polynomials are continuous everywhere
2Where is f(x)=1/(x-4) discontinuous?
x = 4
3Is f(x)=x^2 continuous at x=0?
Yes
4Name the three conditions required for continuity at a point.
f(a) defined, limit exists, limit equals f(a)
5What type of discontinuity has a 'hole' that can be patched?
Removable discontinuity
📄 Continuity — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.