Exercise 7.5.3

Question

The hypothesis in Theorem 7.5.1 (i) that $F^{\prime}(x)=f(x)$ for all $x \in[a, b]$ is slightly stronger than it needs to be. Carefully read the proof and state exactly what needs to be assumed with regard to the relationship between $f$ and $F$ for the proof to be valid.

Ulisse Mini · Accepted Answer

We use the Mean Value Theorem to ensure that for all subintervals of a particular partition $[x_{k-1}, x_k]$, we have $F(x_k) - F(x_{k-1}) = F'(t_k) (x_k - x_{k-1})$. But the MVT does not require that $F$ is differentiable at the endpoints, so we can relax the requirement slightly and have $F'(x) = f(x)$ over the open interval $(a,b)$.

Exercise 7.5.3

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Exercise 7.5.3

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