Question

[3] 10. Prove the theorem that states: If f is differentiable at a, then f is continuous at a. Solution: Assume lim (*)={@) = f'(a). Then, since lim (1 – a) = 0, we get by the Limit Laws that f(x) - - @ -a) = lim + f(x)-f(a) T-a lim( 1a -a) = f'(a)-0 = 0 ma a limf (x) - f(a)] = lim- rta Now, observe that lim f(1) = lim[f (2) - f(a) + f(a)] = lim(f(x) – f(a)] + lim f(a) = 0 + f(a) = f(a) Hence, f(x) is continuous at a.