Question

If f has a discontinuity at 0, then \(\int\limits_{ - 1}^1 {f(x)dx} \) does not exist.

Short answer

The given statement is FALSE.

Step-by-step solution

Step 1: Piecewise function

Consider this piecewise function that has a jump discontinuity at \(x = 0\)

\(f(x) = \left\{ \begin{array}{l}2\;\;\;if\;x < 0\\4\;\;if\;\;x \ge 0\end{array} \right.\)

Step 2: The integral and the graph

The integral exists and in this case equals \(2 + 4\)

\(\begin{array}{l}\int\limits_{ - 1}^1 {f(x)dx} = \int\limits_{ - 1}^0 {2dx + \int\limits_0^1 {4dx} } \\ = 2 + 4\\ = 6\end{array}\)