Show that $\displaystyle f(x)$ is the derivative of$\displaystyle f(.)$ at$\displaystyle x$ if and only if $\displaystyle lim_{h \to 0} \sup_{|t|\leqslant h} \frac{|f(x+t)-f(x)-tf^{'} (x)|}{h} = 0 $
Show that $\displaystyle f(x)$ is the derivative of$\displaystyle f(.)$ at$\displaystyle x$ if and only if $\displaystyle lim_{h \to 0} \sup_{|t|\leqslant h} \frac{|f(x+t)-f(x)-tf^{'} (x)|}{h} = 0 $