Some questions on the connections between continuity and differentiability. Important for your future, regardless of where life takes you.

1) What criteria must be met for a function to be considered continuous at a point where x=a?

2) What does it mean for a function to be considered differentiable at a point where x=a?

3) If a function is continuous everywhere, will it always be differentiable everywhere?

4) If a function is differentiable everywhere, will it always be continuous everywhere?

5) If a function is differentiable everywhere, will its derivative always be continuous everywhere?

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 consideredcontinuousat a point wherex=a?Note that this requirement implies that

f(a) exists.2) What does it mean for a function to be considereddifferentiableat a point wherex=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=arequires that the limit of secant slopes aboutaexists. This limit (the definition of the derivative) requires the existence off(a) and continuity ata.Think about it...how could you evaluate the limit of secant slopes (derivative) aboutaif there was no point ataor if the function was not continuous ata(you need to approachafrom the left and right...right?). Therefore, if you can find the derivative ata, you must have continuity ata.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!

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