Exercise 8.1.12

Question

For each $c \in [a,b]$, explain why there exists a $\delta(c) > 0$ (a $\delta > 0$ depending on $c$) such that $$\left|\frac{F(x) - F(c)}{x-c} - f(c)\right| < \epsilon$$ for all $0 < |x - c| < \delta(c)$.

Ulisse Mini · Accepted Answer

Since $F' = f$, this is essentially the limit defining $F'(c) = f(c)$. So just use the same $\delta$ as that used to define the derivative of $F$.

Exercise 8.1.12

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Exercise 8.1.12

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