For questions involving the concept of uniform continuity, that is, "the $\delta$ in the definition is independent of the considered point".

Continuity of a function is purely a local property, whereas uniform continuity is a global property that applies over the whole space.

Definition:

A real-valued function $~f~$ defined on a set $~E~$ of real numbers is said to be uniformly continuous provided for each $\epsilon~\gt ~0~$, there is a $~\delta~\gt~0~$ such that for all $~x,~x'~\in~E~$,

if$~\quad |x-x'|\lt \delta~$, then $\quad |f(x)-f(x')|\lt~\epsilon~$.

${}$

A mapping from a metric space $~(X,\rho)~$ to a metric space $~(Y,\rho)~$ is said to be uniformly continuous, provided for every $\epsilon~\gt ~0~$, there is a $~\delta~\gt~0~$ such that for all $~u,~v~\in~E~$,

if$~\quad \rho(u,~v)\lt \delta~$, then $\quad \sigma (f(x)-f(x'))\lt~\epsilon~$.

  • uniform continuous $\implies$ continuous, but converse is not true.

e.g., The function $~f(x) = \frac{1}{x}~$ is continuous on $~(0,1)~$ but not uniformly continuous.

  • The $~\delta~$ here depends on $~\epsilon~$ and on $~f~$ but that it is entirely independent of the points $~x~$ and $~y~$. In this way, uniform continuity is stronger than continuity and so it follows immediately that every uniformly continuous function is continuous.

Reference:

https://en.wikipedia.org/wiki/Uniform_continuity#Generalization_to_uniform_spaces

history | excerpt history