What is difference between Discontinuous and not continuous of a function $f$ at some point $x=a?$ I think there is not difference between the two. Is there some difference between these two related to limit points of the domain of the function $f?$ Please describe if there is some difference between Discontinuous and not continuous at some point. Thanks!
2 Answers
These are the same, but there are several kind of discontinuities. Maybe that's the distinction you wanted to make. See for example:
Many authors reserve the word discontinuous to be used only for points where the function is defined (and likewise for continuous, but that's obvious). In that case there is a difference: at a point which is not in the domain, the function can be said to be both "not continuous" and "not discontinuous"! See the answers here: Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large \frac{1}{z}$ by definition discontinuous at $0$?
