# Pathological Derivatives

Pathological Derivatives

This Demonstration explores how the definition of a derivative plays out for some pathological functions. By changing , you can see that in most of the examples, the slopes of the secant lines do not have a finite limit as . An interesting comparison, however, is one between the continuous extensions of and sin(1/x); both are continuous with oscillatory behavior, but only the latter is differentiable at .

t

t→0

xsin(1/x)

2

x

x=0