I'm sure if MacAlpine wasn't banned, he would have replied... Anyway, here are some answers:
1) What criteria must be met for a function to be considered continuous at a point where x=a?
Note that this requirement implies that f(a) exists.
2) What does it mean for a function to be considered differentiable at a point where x=a?
Note that this requirement implies that f(a) exists.
3) If a function is continuous everywhere, will it always be differentiable everywhere?
Not necessarily. Consider functions that are continuous everywhere, but have corners, cusps, or vertical tangents. These functions would not be differentiable at such points.
4) If a function is differentiable everywhere, will it always be continuous everywhere?
Yes. We'll prove this formally in class, but for now... Differentiability at x=a requires that the limit of secant slopes about a exists. This limit (the definition of the derivative) requires the existence of f(a) and continuity at a. Think about it...how could you evaluate the limit of secant slopes (derivative) about a if there was no point at a or if the function was not continuous at a (you need to approach a from the left and right...right?). Therefore, if you can find the derivative at a, you must have continuity at a.
5) If a function is differentiable everywhere, will its derivative always be continuous everywhere?
This is a thinker, but believe it or not...no! Here's an example that utilizes some techniques we'll study in the near future. Watch at your own risk!
Computing derivative as difference quotient: necessary for some piecewise definitions
or
Does Derivative Have to be Continuous? (feat. x^2sin(1/x))