Limits and continuous functions

Question

I have always been told that if $f(x)$ is a continuous function at $a$ so that $f(a) = L$, then $\lim_{x\to a}f(x) = L$. Please, could someone explain in detail why this is true?

Answer 1

It is true because of the way we define "continuous" and because of the meaning of $\lim_{x\to a}f(x)=L.$ Which parts of those definitions are you having trouble with?

Ultimately, most courses will define what they mean by $$\lim_{x→a} f(x)=L,$$ then define "$f(x)$ is continuous at $a$" by in effect saying $\lim_{x→a}f(x)=f(a)$ (in whatever form). There's not much more to it than that.

Answer 2

There really isn’t anything to explain: it’s essentially the definition of continuity of $f$ at $a$.

Definition. Let $f$ be a real-valued function defined on some domain $D\subseteq\Bbb R$, and let $a\in D$ be such that $(a-\epsilon,a+\epsilon)\subseteq D$ for some $\epsilon>0$. (In other words, $D$ contains an open interval around $a$.) Then $f$ is continuous at $a$ if $\lim_{x\to a}f(x)=f(a)$.

The last sentence can be paraphrased as follows to make it look more like your question:

Then $f$ is continuous at $a$ if there is an $L\in\Bbb R$ such that $\lim_{x\to a}f(x)=L$ and $f(a)=L$.

A continuous function is simply a function that is continuous at every point of its domain, so if $f:\Bbb R\to\Bbb R$ is continuous, then by definition $\lim_{x\to a}f(x)=f(a)$ for every $a\in\Bbb R$.

Answer 3

if $f(a) = L$, that means that there are no "holes" in the function and in the graph of the function, and therefore it's continuous.