Prove $|f(y)−f(x)−f′(x)(y−x)|≤ε|y−x|, ∀x,y∈[0,1], |x−y|<δ$. Suppose f is a differentiable function and f′ is continuous on [a,b], 
prove that ∀ε > 0, there is δ > 0 such that
$|f(y)−f(x)−f′(x)(y−x)|≤ε|y−x|, ∀x,y∈[0,1], |x−y|<δ$.
Since f is differentiable and f' is continuous, then by using the definitions of both, I can apply MVT to get to this conclusion, however I am having trouble writing it out so that it can be cohesive.

Apply the triangle inequality and use MVT. Then show that $|f'(c)-f'(x)|\le \epsilon$ for some $c\in(0,1),$ which is true since $f'$ is continuous.
