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
Continuity types classify how a function can fail to be continuous at a point. A removable discontinuity (hole) occurs when the limit exists but doesn't equal f(a). A jump discontinuity occurs when left and right limits differ. An infinite discontinuity occurs when the function approaches ±∞.
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.