Continuity on an Interval
Now that we've got the idea of a continuity at a point down, we can talk about what it means for a function to be continuous on an entire interval.
It shouldn't come as much of a surprise that we say a function f is continuous on the open interval (a,b) if f is continuous at every point c in (a,b), not including the points a and b.
In other words, f is continuous at every point in the interval (a, b).
If we try to graph a continuous function on this interval, we'll be able to draw a nice smooth curve from a to b without ever taking our pencil off the paper. We're going from Point A to Point B completely uninterrupted.