Pathological Derivatives

t

-1

zoom

1

function

Absolute Value

This Demonstration explores how the definition of a derivative plays out for some pathological functions. By changing

t

, you can see that in most of the examples, the slopes of the secant lines do not have a finite limit as

t→0

. An interesting comparison, however, is one between the continuous extensions of

xsin(1/x)

and

2

x

sin(1/x)

; both are continuous with oscillatory behavior, but only the latter is differentiable at

x=0

.