Question

if limx->a f(x)=undef. and f(a)=undef. is f(x) continuous at a? can you have undef limit is that same as does not exist

Answer 1

Undefined is where a determination cannot be made about that particular point.
by definition, both left and right hand limits must exist and be equal for the limit to exist. It doesn't matter what happens at x =a, just values that are very close to a.

Answer 2

For example, if \[\lim_{x \rightarrow a}f(x) = \frac{ 0 }{0}\] doesn't necessarily mean it doesn't exist or is undefined, it just means it needs a little more work.