I've been studying real analysis over the past few months, and I'm having trouble organizing the different notions of continuity and ideas related to continuity in my head geometrically. I will explain my notions in general terms (in other words, without delta-epsilon definitions). Can you tell me how to sharpen my intuitive thinking wherever my ideas are incorrect or too general?
Continuous - The preimage of every open set is an open set.
Lipshitz Continuous - The absolute values of the slopes of all secant lines for the function are bounded.
Uniformly Continuous - The best notion I have is that it's just what a continuous function over a closed and bounded set is. One idea I have is that it means the function is the uniform limit of some series of piecewise linear functions, but does this hold for uniformly continuous functions over domains that are not compact?
Absolutely Continuous - This is my shakiest notion geometrically. The delta-epsilon definition gives me a loose notion of being able to break up the condition for uniform continuity over disconnected unions of open intervals. I know absolutely continuous functions have to be of bounded variation, which carries a geometrical notion of a continuous function whose image over any partition of the domain has a finite arc length, but I can't see what makes this notion stronger visually, nor can I grasp how it connects so well to the Fundamental Theorem of Calculus.
Another question: how do these different types relate to one another, and what examples can show these relations? I know the Cantor function is a good example of a uniformly continuous function (one of bounded variation) that is not absolutely continuous, but is it Lipschitz? If not, is there a function that is Lipschitz but not absolutely continuous?
I appreciate your input, and I apologize if this question is too general and lacking in rigor - I am still learning my way around this site!