If I understand correctly, the formal definition of a continuous function $f:X \to Y$ is when
$$ \lim_{x \to c}f(x)=f(c) $$
Where $c$ is a constant, and $\{x,c\} \in X$. The layman explanation I was given by my schoolteacher was that a function is continuous if you can draw the graph without your pen going off the page. I only know of one exception to this intuition:
$f(x)=\frac{1}{x}$ has a point of discontinuity at $x=0$, but is still a continuous function because it has no discontinuities in its range:
$X=\mathbb R-\{0\}$
(Side question: am I misusing the equals sign in the above expression?)
Are there other examples of a:
- Continuous function where you must take your pen off the page?
- Discontinuous function where you don't have to take your pen off the page?
(Please note that for question $1$, I'd be more interested in examples that aren't similar to the one I have already given.)
Thank you for reading.