Question

Hello, just joining up. Prof. Jerrison is very quick and engaging. I have a question about the proof that differentiability = continuity in lecture 2 (https://itunes.apple.com/us/podcast/lecture-01-derivatives-slope/id354869053?i=80690404) --- did I miss where it was shown that (the limit as x->z of a*b) is equal to (the limit as x->z of a)*(the limit as x->z of b)? The proof seems to hinge on that. It's certainly an intuitive inference, but is there a simple proof of said inference?

btw, the proof is the last few minutes of lecture 2 in the link referenced above.

Answer 1

No you didn't miss it. The lectures did not cover limit laws and I don't think they will. Later the professor shows a geometric proof of sin x / x but doesn't use the squeeze theorem. I guess 18.01 isn't too concerned with rigorous proofs but more about giving us an intuition. It's somewhat annoying to me, but I guess we need to trust the process and if we're concerned with more rigor at this stage, we must go outside the class.

Btw, I looked for that differentiability implies continuity and found much more indepth proofs such as : http://people.hofstra.edu/stefan_Waner/RealWorld/calctopic1/contanddiffb.html at the bottom of the page.

Answer 2

Also check out Albert Strang's Highlights of Calculus, I found it really useful: http://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/index.htm

Answer 3

thanks for your replies guys, very helpful!