Question

prove that if a function is derivable at a point, then it is cuntinuous at that point

Answer 1

\(\underset{x\rightarrow c}{lim}\frac{f(x)-f(c)}{x-c}\) = f'(c)

\(\frac{\underset{x\rightarrow c}{lim}\,f(x)-f(c)}{\underset{x\rightarrow c}{lim}\,x-c}\) = f'(c)

\(\underset{x\rightarrow c}{lim}\,f(x)-f(c)\) = f'(c) . \(\underset{x\rightarrow c}{lim}\,(x-c)\)

\(\underset{x\rightarrow c}{lim}\,(f(x)-f(c))\) = f'(c) . (c - c)

\(\underset{x\rightarrow c}{lim}\,f(x)\) - \(\underset{x\rightarrow c}{lim}\,f(c)\) = f'(c).(0)

\(\underset{x\rightarrow c}{lim}\,f(x)-f(c)\) = 0

\(\underset{x\rightarrow c}{lim}\,f(x)\) = f(c)