All numbers discussed here is in $$\mathbb{R}$$

What I’d like to discuss here is all about uniform concepts I’ve learnt since. And I wanna summarize them.

In calculus, there exists a concept called Continuity as expressed below: \begin{aligned}&\mbox{f(x) is continuous at x_0}\Leftrightarrow\\ &\forall{\varepsilon\in \mathbb R}\exists{\delta\in \mathbb R} : \vert x-x_0\vert <\delta \Rightarrow \vert f(x)-f(x_0)\vert <\varepsilon\end{aligned}

As is seen above, $$\delta$$ is somewhere dependent towards $$x_0$$. What if we genernalize that idea such that $$\delta$$ shall be dependent not to $$x_0$$ but to an interval? Just modify the concept as below: \begin{aligned}&\mbox{f(x) is \textbf{uniform} continuous at interval \mathbf{I}}\Leftrightarrow\\ &\forall{\varepsilon\in\mathbb R}\exists{\delta\in\mathbb R, x_1,x_2\in\mathbf{I}}: \vert x_1-x_2\vert <\delta \Rightarrow \vert f(x_1)-f(x_2)<\varepsilon\vert \end{aligned}

And with Lagrange’s mean value theorem, we can distinguish that in an open interval, if every point is derivable and its value is bounded ($$\forall x \in\mathbf{I}\exists A \in\mathbb{R}, \vert f'(x)\vert <A$$), that function is then uniform continuous.

Proof.

If $$f(x)$$ is derivable in $$\mathbf{I}$$, according to Lagrange’s mean value theorem, we can conclude that $$\frac{f(x_1)-f(x_2)}{x_1-x_2}=f'(\xi)\leq \max\{f'(x)\}< M$$, thus denoting $$\vert x_1-x_2\vert \leq \delta$$ and we can say $\vert f(x_1)-f(x_2)\vert <M\delta<\varepsilon\Rightarrow \delta=\frac{\varepsilon}{M}$

Just let $$\delta$$ be $$\frac{\varepsilon}{M}$$ and then the requirements for uniform continuity are met.

That type of continuity is particularly called Lipschitz continuity.

### in a prettier writing style

If we combine the limit’s symbols with the continuity concept, the statement can be easily expressed as follows. $\mbox{f(x) is continuous at x_0}\Leftrightarrow\lim_{x\to x_0}f(x)=f(x_0)$ The same way the uniform continuity can be as: \begin{aligned}\mbox{\forall{a,b}\in\mathbf{I}, f(x) is uniform continuous at \mathbf{I}}\Leftrightarrow\lim_{\Vert T\Vert \to 0}\vert f(a)-f(b)\vert =0 \\(\mbox{\Vert T\Vert  denotes \vert a-b\vert }) \end{aligned}