Question

Hello,

I've just started the course and wanted to get some clarification on the Differentiable implies Continuous theorem.

I understand up to the point,

Lim x -> x0 ((f(x) - f(x0))/x - x0)(x - x0)

Then he uses the diff -> cont assumption to state that the first factor here = f'(x0)

I could understand this if the restructuring of the definition of continuity wasn't divided by x-x0, which he states in the second factor = 0???

I'd really appreciate some clarification here.

Thanks heaps.

Answer 1

I'm not entirely certain what you're asking (I'm not taking the course), but it sounds like you're confused by one of the formulas for a derivative:
\[f'(x_0)=\lim_{x\rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}\]
While this obviously looks a little different than the general derivative equation, think about what this is saying: the numerator is finding a smaller and smaller change in y, and the denominator is finding a smaller and smaller change in x until you have an infinitely small change in y at a point over an infinitely small change in x at the same point: hence the derivative of the function at that point.

Answer 2

Thank you mBoorstin, rephrasing it like that has made it click. As an aside, where can i find the syntax for writing equations like that?

Thanks for your help.

Answer 3

That's something called LaTeX. You can either google a tutorial for it, or use the Equation button at the bottom (next to Draw and Attach File) to create it for you.

Answer 4

Great, thanks.