Let f : (a,b) → R and |f′(x)| < M, ∀x ∈ (a,b). Show that |f(x)−f(x′)| ≤ M|x−x′| for all x,x′ ∈(a,b).

I need Help on this as soon as possible

The hypothesis |f '(x)| < M tells us that f is differentiable on (a, b). So let x, and x' be in (a, b). We can assume without loss of generality that x ≤ x'. If x = x', the result is trivial. Otherwise, we can say that f is continuous on [x, x'] (why?) and differentiable on (x, x') so that the Mean Value Theorem ensures that for some c in (x, x')

f(x) - f(x') = f '(c)(x - x').

Using the boundedness condition given, we have

|f(x) - f(x')| = |f '(c)| |x - x'| < M|x - x'|

as required.

Can you explain the (why?)

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How is this math? There arent any numbers!