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?


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.

  • $\begingroup$ What is the meaning of $\lim_{x\to a}f(x)=L$? $\endgroup$ – Lucas Alanis Jun 2 '13 at 4:35
  • 2
    $\begingroup$ @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

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 $a\in\Bbb R$.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.