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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