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Assume $f$ has a domain of $[a, b]$. Is it possible that $f$ is continuous at $x = a$ and $x = b$?

If the definition of continuity is that the left and right limits are equal to the function at the given point, then this fails at $a$ since the left limit is undefined, and fails at $b$ since the right limit is undefined there.

On the other hand there are other questions on this site that imply that it is possible for a function to be continuous on a closed interval, so perhaps this is simply an incorrect definition of continuity that I have seen in some high school text books.

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    $\begingroup$ Your definition of continuity is incorrect. Continuity on a point does not imply that right and left limits are equal: it implies that $IF$ they exist, they are equal. $\endgroup$ – Crostul May 23 '15 at 9:32
  • $\begingroup$ In a Calculus textbook (Stewart), they mention that $f$ is continuous on a closed interval $[a,b]$ when they talk about the mean value theorem. They are referring to continuity from the right (or left) at the endpoint $a$ (or $b$). $\endgroup$ – MathNewbie May 23 '15 at 9:41
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Your definition is correct only for interior points of the domain. For end points of the domain, the limit to be considered is the corresponding one-sided limit.

A couple of references.

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  • $\begingroup$ Can you please give a reference for this, or an explanation for why this should be the case? $\endgroup$ – bryn May 24 '15 at 5:22

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