Analysis I | Part IA, 2001

Explain what it means for a function f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} to be Riemann integrable on [a,b][a, b], and give an example of a bounded function that is not Riemann integrable.

Show each of the following statements is true for continuous functions ff, but false for general Riemann integrable functions ff.

(i) If f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} is such that f(t)0f(t) \geq 0 for all tt in [a,b][a, b] and abf(t)dt=0\int_{a}^{b} f(t) d t=0, then f(t)=0f(t)=0 for all tt in [a,b][a, b].

(ii) atf(x)dx\int_{a}^{t} f(x) d x is differentiable and ddtatf(x)dx=f(t)\frac{d}{d t} \int_{a}^{t} f(x) d x=f(t).

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