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=5x=5 and then stop, is x=5 a point of continuity? I don’t think so because there are no points for x>5x>5.
Would the interval that this function is continuous be (−∞(-\infty,5) or (−∞,5](-\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=5x=5 (but not slightly before), then the function is still continuous at that x=5x=5. Even if you would only draw a dot at x=5x=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.