Suppose that f is a constant function in an interval (a,b) on the x-axis
i.e. f(x) = c ∀ x ∊ (a,b) and f(x+h) = c ∀ x + h ∊ (a,b)
f(x+h) − f(x)
∴ f '(x) = ————— h
∴ f '(x) = ————— h
c − c
= ——— = 0 ∵ h ≠ 0 by definition
h
∴ a Constant Function has a zero derivative= ——— = 0 ∵ h ≠ 0 by definition
h
so the graph of f is a horizontal line