Collegeboard and the AP Calculus exam is done

[mathwonk]

sorry. i mixed two concepts, namely
1) how to characterize a function which is integrable, (one with a measure zeros et of discontinuitties),
and 2) how to characterize a function which is an integral, namely it is lipschitz continuous.

they are unrelated.

 

[mathwonk]

i seem to have made a simple concept look hard. lipschitz continuity is much easier than regular continuity.

i.e. it just says that for some K>0, the change in y is never more than K times the change in x.

this makes it very easy to check regular continuity, namely that small changes in x produce small changes in y.


do you know the epsilon delta definition of continutity? i.e. that for every epsilon, there must exist a delta such that when x changes by less than delta, then y changes by less than epsilon?


well how do you find that delta? say for f = cosine, at pi/6?


the lipschitz princiople tells you to look for a bound on the slope of f, namely 1, in this case, since the derivative is bounded by 1.

thus delta can always be taken as 1.epsilon!


how easy is that? did you do exercises where you were given an epsilon and had to find a delta?

if not you may not appreciate what this does for you, but if you did, you should.

 

[mathwonk]

on the other hand if you had a sophisticated course and learned about sets of measure zero, try this: prove: if f is lipschitz continuous on [a,b] and has derivative equal to zero except on a set of measure zero, then f is constant.