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- Apr 14, 2013

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Let [tex] g: R \rightarrow R [/tex] a bounded function. There is a point [tex] z \epsilon R [/tex] for which the function [tex] h: R [/tex] \ [tex]\{z\} \rightarrow R [/tex], where [tex] h(x)=\frac{g(x)-g(z)}{x-z} [/tex] is not bounded. Show that the function [tex] g [/tex] is not differentiable at the point [tex] z [/tex].

My idea is the following:

Let the function [tex] g [/tex] be differentiable at the point [tex] z [/tex], so [tex] lim_{x \rightarrow z}{ \frac{g(x)-g(z)}{x-z}}=L [/tex]. So there is a ε>0: [tex] |\frac{g(x)-g(z)}{x-z}-L|<ε [/tex].

[tex] |h(x)|=|h(x)-L+L| \leq |h(x)-L|+|L|=|\frac{g(x)-g(z)}{x-z}-L|+|L|<ε+|L| [/tex].

So the function [tex] h[/tex] is bounded. That cannot be true. So the function [tex] g [/tex] is not differentiable at the point [tex] z [/tex].

Is my idea right??