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Proof of the existence of a limit.
I am following the proof of the classical mean value theorem in Rudin’s analysis. In the proof, he stated:
«Suppose that $f$ is continuous at $c$ and that $f^prime (c)$ exists. Let $F(x) = f(c+x) – frac{f(c+x)-f(c)}{x}$. Then $F$ is continuous at $x=0$ and hence $lim_{xrightarrow0}F(x) = F(0) = f^prime (c)$. We have $F(x) – F(0) = xf^prime (c) + o(x)$, so $f^prime(c) = frac{F(x)}{x} + o(frac{1}{x})$. Now $x^{ -1}F(x)$ is a bounded function, so $lim_{xrightarrow infty} x^{ -1}F(x) = lim_{xrightarrow infty} f^prime(c) = 0$.»
I don’t understand why $F(x) = f(c+x) – frac{f(c+x)-f(c)}{x}$. I know that $f(c+x)$ is equal to $f(c)+xf^prime(c)+o(x)$ as $xrightarrow 0$. But how does the expression $-frac{f(c+x)-f(c)}{x}$ match with that?
A:
$$f(c+x) – frac{f(
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