Answer to Question #310128 in Real Analysis for Kaygold

Question

A real function f defined on an interval [a,b] with a<c<b where c is a point of the interval, is said to be differentiable at the point x=c if

Answer 1

A function f is said to be differentiable at x = c, c $\in$ (a,b) if both left hand derivative and right hand derivative at x = c exist finitely and they are equal also. That means

(i) $\lim_{x\to c^{-}}\frac{f(x)-f(c)}{x-c}$ exists finitely

(ii) $\lim_{x\to c^{+}}\frac{f(x)-f(c)}{x-c}$ exists finitely

and (iii)

$\lim_{x\to c^{-}}\frac{f(x)-f(c)}{x-c}$=$\lim_{x\to c^{+}}\frac{f(x)-f(c)}{x-c}$