WOLFRAM|DEMONSTRATIONS PROJECT

Pathological Derivatives

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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
.