0
$\begingroup$

This is a question from a calculus sample test, and I can't figure out how to prove it. Can I get some help from you guys?

Definition of continuity that we've learned is $$\lim_{x\to a} f(x) = f(a).$$ If that holds, then $f$ is continuous at $a$.

The definition that we learned of a limit is:

For every $\epsilon > 0$, there exists $\delta > 0$ such that $|x - a| < \delta$ implies $|f(x) - L| < \epsilon$.

$\epsilon$ and $\delta$, as far as I can tell, are just variables, $a$ is what $x$ is approaching, and $L$ is the limit.

  • 0
    Is your definition of continuity at $a$ that $\lim_{x \to a} f(x) = f(a)$? You should mention this, as well as your question and what you've tried, in the body of the post for better results. I do think you'll get some excellent answers any moment now, though.2011-10-01
  • 1
    Yes, this sort of theorem depends on your definitions,and usually follows straight from them. In particular, it depends on the definition of "limit" and the definition of "continuous at a.". Different books use different definitions, so it is hard to give a proof without these definitions.2011-10-01
  • 0
    OK. What you need to show is $\lim_{x\to a}f(x)=f(a)$ *if and only if* $\lim_{h\to 0}f(a+h)=f(a)$. Do you know what does "if and only if" mean? Furthermore, do you know the definitions of these two limits?2011-10-01
  • 0
    I know what if and only if means in English, and I think it means something very similar in math. I'm editing in the definitions we learned now.2011-10-01
  • 0
    @Lman: Even "if" alone means something subtly different in mathematics than it does in English. For example, "Paris is the capital of France **if** $x^2<0$" is close to nonsense in ordinary English, but in math it is meaningful and true.2011-10-01

4 Answers 4