Let $X, Y, Z$ be metric spaces. $f: X \rightarrow Y$ is called uniformly continuous if
$\forall \epsilon 0 \quad \exists \delta 0 \quad \forall x, y \in X : d_{x}(x,y) \leq \delta \Rightarrow d_{Y}(f(x),f(y))\leq \epsilon $
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b) $f(x):=x^{2}$ is continuous, but not uniformly continuous on $\mathbb{R}$

Question

Let $X, Y, Z$ be metric spaces. $f: X \rightarrow Y$ is called uniformly continuous if
     $\forall \epsilon > 0 \quad \exists \delta > 0 \quad \forall x, y \in X : d_{x}(x,y) \leq \delta \Rightarrow d_{Y}(f(x),f(y))\leq \epsilon $
   Show:

   b) $f(x):=x^{2}$ is continuous, but not uniformly continuous on $\mathbb{R}$

zarkon · Answer

your questions are fairly easy, but if you want help from people who know how to do them then you should show some effort on your part.

anonymous · Answer

what kind of effort? the problem is i  am not so good at math and i need to deliver my mentor about in 4 hours this and some additional questions with an answer, if not i wont be able to participate  in about one and half month at final exam...