Is it true that if f is continuous, lim h → 0 1 h ∫...

ziphumulegn

ziphumulegn

Answered

2022-07-02

Is it true that if f is continuous, lim h 0 1 h x x + h f ( t ) d t = f ( x )

Answer & Explanation

Alexia Hart

Alexia Hart

Expert

2022-07-03Added 19 answers

There is no need to invke the FTC for this. Note that, by the monotonicity of the integral,
h min x s x + h f ( s ) = x x + h min x s x + h f ( s ) d t x x + h f ( t ) d t
Similarly,
h max x s x + h f ( s ) = x x + h max x s x + h f ( s ) d t x x + h f ( t ) d t
Now divide these inequalities by h and let h 0. By continuity,
lim h 0 min x s x + h f ( s ) = f ( x )
with a similar result for the max. Combinining the resulting inequalities yields the result.

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