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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?

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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.

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  • $\begingroup$ What is the meaning of $\lim_{x\to a}f(x)=L$? $\endgroup$ – Lucas Alanis Jun 2 '13 at 4:35
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    $\begingroup$ @Lance: There are several ways of saying what we mean by $\lim_{x\to a}f(x)=L$. This may be done by an $\epsilon,\delta$ definition, by an open sets definition, by a sequence limit definition, a function limit definition, or several other ways. How was it done for you? $\endgroup$ – Cameron Buie Jun 2 '13 at 4:41
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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 $a\in\Bbb R$.

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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.

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