1. The problem statement, all variables and given/known data “Let f ′ exist on (a, b) and let c ∈ (a, b) . If c + h ∈ (a, b) then (f (c + h) − f (c))/h = f ′(c+θh). Let h→0 ; then f ′ (c + θh) → f ′ (c) . Thus f ′ is continuous at c .” Is this argument correct? 3. The attempt at a solution I'm pretty sure the argument's wrong - is the flaw simply that in saying f ′ (c + θh) → f ′ (c) you have assumed the result that f' is continuous at c in proving the result? I.e proving lim f'(x) as x→c=f'(c) by assuming lim f'(x) as x→c=f'(c)? Or is there a different error in the statement?