First derivative test fails for function that is discontinuous at the critical point
Suppose is a function and is a point in the interior of the domain of . Suppose is not continuous at . Then, we cannot use the first derivative test directly, and naively looking at the statement of the test could yield incorrect conclusions.
Consider the function:
We see that:
Note that is not defined at 0 because is not continuous at zero.
We thus see that for to the immediate left of 0 and for to the immediate right of 0. Thus, is changing sign from negative to positive. A naive application of the first derivative test suggests that has a local minimum at zero. However, this is not correct. The reason is that the function jumps upward at 0, hence it does not attain a local minimum from the left at zero.