Practice Types of Continuity and Discontinuity in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
A function is continuous at x = a if \lim_{x \to a} f(x) = f(a). Discontinuities are classified as removable (limit exists but doesn't equal f(a)), jump (left and right limits exist but differ), or infinite (function blows up to \pm\infty).
Continuous means you can draw the graph without lifting your pen. A removable discontinuity is a single hole you could fill in. A jump discontinuity is a gap where the function leaps to a different value. An infinite discontinuity is where the function shoots off to infinity (a vertical asymptote).
Example 1
easyClassify the discontinuity of f(x) = \dfrac{x^2 - 4}{x - 2} at x = 2.
Example 2
mediumDetermine whether the piecewise function f(x) = \begin{cases} x^2 & x < 1 \\ 3 - x & x \geq 1 \end{cases} is continuous at x = 1.
Example 3
easyClassify the discontinuity of g(x) = \frac{1}{(x-3)^2} at x = 3.
Example 4
mediumFind the value of c that makes h(x) = \begin{cases} cx + 1 & x \leq 2 \\ x^2 - 1 & x > 2 \end{cases} continuous at x = 2.