= Epsilon-delta definition of continuity
{title2=$\forall\varepsilon>0\ \exists\delta>0$}
A <function> $f:D\to\mathbb R$ is <continuous> at $a\in D$ when
$$
\text{for every }\varepsilon>0\text{ there is a }\delta>0\text{ such that}
\quad x\in D,\ |x-a|<\delta\ \Longrightarrow\ |f(x)-f(a)|<\varepsilon.
$$
First choose the required output tolerance $\varepsilon$; then choose an input tolerance $\delta$ that works for every allowed <function input> in that window. Picking a different $\delta$ separately after seeing each $x$ would reverse the <quantifiers> and lose this guarantee. There is no need to exclude $x=a$, because the output difference is then zero.
At an isolated point of the <domain of a function>, choose a small enough window that its only allowed <function input> is $a$. This proves <continuity> there directly. At a point with arbitrarily close other inputs, the same definition is the <limit of a function> condition $f(x)\to f(a)$. For <metric spaces>, replace the two <absolute values> by their <distance functions>; for arbitrary <topological spaces>, the corresponding definition is a <continuous map>.
The input tolerance may depend on $a$. A single choice valid everywhere on the <domain of a function> is the stronger requirement of <uniform continuity>. The <continuous function> $x\mapsto x^2$ on $\mathbb R$ shows the difference: for any proposed $\delta>0$, the points $a=1/\delta$ and $b=a+\delta/2$ have $|b-a|<\delta$, yet $b^2-a^2=1+\delta^2/4>1$. Thus the fixed output tolerance $\varepsilon=1$ admits no global input tolerance on the <real line>.
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