Examples 7.1.20(c):

Show that if one starts with an integrable function f in the Fundamental Theorem of Calculus that is not continuous, the corresponding function F may not be differentiable.
Back

Our Fundamental Theorem states that if we start with a continuous function f(t) over some interval [a, b] then the new function F(x) obtained by integrating f from a to some variable value x is differentiable:

F(x) = f(t) dt is differentiable as long as f is continuous

Now let's start with a simple step function that is integrable but not continuous over the interval, say, [-1, 1]. Define

Then for if x < 0 we have:

and for x 0

But then

which is not differentiable at x = 0.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007