Proving $ f(x) = x^2 $ is not uniformly continuous on the real line

It is continuous. However, it is not uniformly continuous.

Suppose it were; then for every $\epsilon$, there exists a $\delta$ for which $$|x - y| < \delta \implies |x^2 - y^2| < \epsilon$$

However, consider $\epsilon = 1$; if such $\delta$ existed and $y = x + \frac{\delta}{2}$, we would find that

$$|x^2 - (x + \frac{\delta}{2})^2| < 1$$

for every real $x$; however, this would imply that $$|x \delta + \frac{\delta^2}{4}| < 1$$ which is a clear contradiction, since we can choose $x$ large.

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. ∎

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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 $\mathbb{R}.$