What does it mean for a function to be continuous on its domain? 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?

A function is said to be continuous if it continues at each point. This means that over the domain. Functions that are not continuous do not exist for every x value over the domain. For example if a function is defined near an open interval (the circle that is not shaded on a graph) then the function is discontinuous. However, if a function is defined near a closed circle (the shaded circle of a graph) then that function is continuous. This problem usually occurs when trying to find the Domain and Range of a function or if a problem ask to graph the Domain and Range of a function.
