Can we express uniform continuity in terms of uniform convergence?

Question

Is it true that $f$ is uniformly continuous if $f(z) \to f(z_0)$ uniformly as $z \to z_0$?

And I am a bit confused why all the definitions treat only the uniform convergence of sequences, why can't we let the index to be continuous.

If not, is there a way to express uniform continuity in terms of uniform convergence?

Answer 1

We say that a function $f$ is uniformly continuous on a metric space $X$ if for all $\epsilon>0$ there exists $\delta>0$ such that for all $x,y \in X$, $d(f(x),f(y))<\epsilon$ whenever $d(x,y)<\delta$.

Answer 2

$f_n(x)=n$ on $[-1, 1]$ it is uniformly continuous since it has bounded derivation but not uniformly convergence (for given $\epsilon$ let $x=(2\epsilon)^{\frac{1}{n}}). $

Note that: theorem: uniform convergence of a sequence of functions imply uniform continuity.