Suppose that f is a real-valued function on R whose derivative exists at each point and is bounded. Prove that f is uniformly continuous.
Answer
Since f′ is bounded then there's M>0 s.t.
|f′(x)|≤M∀x∈R
hence by mean value theorem we find
|f(x)−f(y)|≤M|x−y|∀x,y∈R
so f is a lipschitzian function on R and therefore it's s uniformly continuous on R.
No comments:
Post a Comment