If I draw a graph of a function and put my pen on the paper and draw a line from negative infinity up until and including the point at $x=5$ and then stop, is x=5 a point of continuity? I don't think so because there are no points for $x>5$.
Would the interval that this function is continuous be $(-\infty$,5) or $(-\infty,5]$?
A function does not necessarily have to be defined around a point, in order to be continuous at the point. So if you stop drawing precisely at $x=5$ (but not slightly before), then the function is still continuous at that $x=5$. Even if you would only draw a dot at $x=5$ and nothing else, then still the function would be continuous there! The only important thing is that the function does not make a jump.
