Proving $ f(x) = x^2 $ is not uniformly continuous on the real line This is  homework problem and the very premise has me stumped. It's in a text on PDE. 
The exercise says to show that $ f(x) = x^2 $ is not uniformly continuous on the real line. But every definition I know says that it is a continuous function, and unless you attach some special condition, like restricting the interval or making it a periodic function (perhaps saying $f(x-2) = f(x)$ or some such) it's by definition continuous. There's always a derivative since $f'(x) = 2x$. 
The preceding chapter is about Drichelet and the like, as an extension of Fourier series, so I am guessing that a Fourier expansion does something here but every proof of the proposition seems to have nothing to do with Fourier series in the slightest. 
So I am pretty lost here. This whole question seems utterly nonsensical. 

Here is a more general approach to solving this problem.
Let me formulate and prove a theorem:

Theorem: 
Let $E = [a,+\infty),$ function $f:E \rightarrow \mathbb{R}$ is
  differentiable on $E$ and $$\displaystyle{\lim_{x \to \infty}} f'(x) =
\infty.$$ Then $f$ is not a uniformly continuous function.

Proof: 
Let the function $f$ be uniformly continuous. Let $\epsilon = 1$ and $\delta>0$ satisfies the deffinition of uniform continuous. From the $\displaystyle{\lim_{x \to \infty}} f'(x) = \infty$ follows that $$\exists m\geqslant a: \forall x \geqslant m \\ \left|f'(x)\right| > \frac{2}{\delta}$$
Let $x_1=m, x_2=m+\frac{\delta}{2}.$ Using Lagrange theorem for $[x_1,x_2]$ we have $$\left|f(x_1)-f(x_2)\right| = \frac{\delta}{2}\left|f'(\zeta)\right| \\ \text{for some } \zeta \in[x_1,x_2] $$
Since $\zeta \geqslant m$ then $\left|f'(\zeta)\right|>\frac{2}{\delta}$ from where $$\left| f(x_1) - f(x_2)\right| > 1 = \epsilon.$$
This contradicts the uniform continuity of the function $f$. Thus, $f$ is not uniformly continuous. 
∎

So, your function is $f(x)=x^2$. Its derivative is 
$$\frac{d(x^2)}{dx} = 2x.$$
The limit of this derivative at infinity is 
$$\displaystyle{\lim_{x \to \infty}} (2x) = \infty.$$
So, by the theorem this function is not uniformly continuous on the $[0,+\infty)$. The same arguments can be used for $(-\infty,0]$. And the statement that if a function is uniformly continuous on the [a, c] as it is on [c,b], then it is uniformly continuous on the [a, b] finally can be used to prove that your function is not uniformly continuous on the real line $\mathbb{R}.$
