Finding example of a function having the required property.

Question

Does there exist any continuous function $f : \Bbb R \longrightarrow \Bbb R$ which is not differentiable only at the integers and is not uniformly continuous everywhere?

The only function which I can think of is the periodic extension of the modulus function on $[-1,1]$ or something similar to that which are uniformly continuous everywhere. Hence they don't suit my purpose. Can anybody help me in finding out an example which satisfies the above condition?

Thank you very much.

Answer 1

Let $f$ be any function which is differentiable at $x$ iff $x$ is not an integer. [There are many such functions and you can build one using $|x|$]. If $f$ is not uniformly continuous we are done. If it is uniformly continuous then $f(x)+x^{2}$ satisfies your requirements.

Answer 2

For one example, define $f(n) = n^2$ for each integer $n$ and then $f(x)$ is linear on $n \leq x \leq n+1.$ In one line, this can be written as $$ f(x) = [x]^2(1-\{x\}) + (1+[x])^2\{x\}$$ where $[x]$ is the floor and $\{x\}$ is the fractional part.