For this graph here, would you say it's strictly increasing over the interval (0,1) and (3,5) OR [0,1] and [3,5]? I'm just a bit confused about whether we include cusps and endpoints.
thanks
Note that definition of strictly increasing/decreasing has NOTHING to do with the derivative - and as a result, you DO include end-points.
A function is strictly increasing over an interval if for every b>a on that interval, f(b)>f(a). Let's consider the point x=1 - in this case, b=1, and f(b)=1. Pick whatever you want for a. If a=0.5, f(a)=0.5<f(b). We can include 0.5 in that interval. In fact, no matter how close a gets to x=1 (or the limit as a goes to 1), f(b) is always going to be bigger. "But keltingmeith", you say, "this explanation means it would be an OPEN interval, not a closed one!" And the answer to that is it doesn't make sense to check for when a=b - because of course f(b) is not going to be bigger than f(b), they must be the same. As a result, the mathematical community basically just decided that it's okay to include end-points, and so that's what people do. Kind of frustrating explanation, I'm sure, but it is what it is.