Question 1185676: Show that if f’(x) is a piecewise continuous on [a,b] , then f(x) is piecewise continuous as well.
Found 2 solutions by Alan3354, robertb:
Answer by Alan3354(69443)   (Show Source):
Answer by robertb(5830)  (Show Source):
You can put this solution on YOUR website! A piecewise continuous function is defined as a function that has a finite number of discontinuities
and doesn't blow up to infinity anywhere.
Let  and  be two such discontinuities of the function inside the interval [a,b].
Then f'(x) is continuous over the open interval (c,d), and hence exists over that same interval.
By theorem, if f'( ) exists implies that  is continuous. Hence f(x) is continuous for every over the interval (c,d).
===> the set of discontinuities for f(x) is a subset of the set of discontinuities of f'(x).
Also, since f'(x) is bounded (due to piecewise continuity), f(x) will also be bounded.
(Otherwise, f(x) will have an infinite discontinuity, contrary to hypothesis.)
Therefore f(x) is also piecewise continuous over [a,b].
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