I never understood the phrase "continuous on its domain."
Isn't everything continuous on its own domain, since the domain are all the $x$ values that we can plug into $f(x)$ and get a defined $y$ value back? i.e. doesn't the domain by definition tell you where the function is continuous? Why would the domain ever include something not continuous / not defined?
Best Answer
Not necessarily. The domain of a function tells you over what values the function $f(x)$ exists, not where it is continuous. Take the piecewise function:
$$f(x) = \begin{cases} 1 & x<0\\ 2 & x\geq0 \end{cases}$$
This function is defined for all $x\in\mathbb{R}$, but is not continuous at $x=0$. It still has a valid value: $f(0)=2$, but that doesn't make it continuous at that point.
For a function to be continuous at a point, its limit must be the same regardless of what direction of approach. In this case, $\lim\limits_{x\to0^-}{f(x)}=1$ while $\lim\limits_{x\to0^+}{f(x)}=2$, making it discontinuous at that point.