with your kind encouragement I have tried typing the post into a wqord document and pasting:
third try:
the two fundamental theorems of calculus are about the relation betwen two ideas: integrals and antiderivatives. Contrary to popular belief they are not the same, but are closely related. or rather for them to be the same in general, one must define antiderivative more carefully than is usually done.
basically the first FTC says to what extent an integral is also an antiderivative, and the second FTC says to what extent an antiderivative is also an integral.
For example, for continuous functions f, antiderivatives and integrals are essentially the same, but why restrict the discussion to continuous functions, when many other functions also have integrals? I.e. to what extent is the integral of a non continuous function also an antiderivative?, and to what extent is the antiderivative of a non continuous function also an integral?
The usual theorems are:
1) If f is continuous on [a,b] then for every x in [a,b] f is also Riemann integrable on [a,x] and the function F defined by F(x) = Riemann integral of f on [a,x], is an antiderivative of f on [a,b]. I.e. F’(x) = f(x) for all x in [a,b].
2) If f is continuous on [a,b], and if G is any diferentiable function on [a,b] such that G’(x) = f(x) for all x in [a,b], then G differs from the previously defined F by a constant, i.e. G(x) -G(a) = F(x), for all x in [a,b].
The proof of the second theorem is obtained by combining the first theorem with the mean value theorem, i.e. given the first theorem, the second one is really the mean value theorem, which tells you the relation between two functions having the same derivative.
But the theorems are more interesting if f is only assumed Riemann integrable but not continuous. We say a statement about [a,b] is true almost everywhere, if for each e>0, the set where it is not true can be covered by a sequence of intervals of total length less than e.
1)A If f is Riemann integrable on [a,b], then f is continuous almost everywhere, hence the function F(x) = integral of f on [a,x], is differentiable with derivative equal to f almost everywhere.
(This is due to Riemann.)
Suprisingly, it is NOT true that any function G which is differentiable with derivative equal to f almost everywhere on [a,b] differs from F by a constant, not even if we assume G is continuous. Namely there exists a (Cantor) function G continuous on [a,b] and with derivative equal to zero almost everywhere, and yet with G not only not constant, but with G increasing weakly monotonically from 0 to 1.
Thus if we take f to be a function which equals zero at every point where G has derivative zero, but f=1 elsewhere, then the integral of f is zero on [a,b], but the antiderivative G of f is not constant, hence does not differ from F by a constant. I.e. the mean value theorem fails for functions like G which are only differentiable almost everywhere, and hence the second FTC, which is essentially the mean value theorem, also fails in this form.
However, we can recover a version of the mean value theorm, hence a second FTC by introducing the concept of Lipschitz continuity. G is Lipschitz continuous on [a,b] if there is some constant K such that for all x,y in [a,b], we always have |G(y)-G(x)| less than or equal to K|y-x|.
1)B If f is Riemann integrable on [a,b], then f is continuous almost everywhere, hence the function F(x) = integral of f on [a,x], is differentiable with derivative equal to f almost everywhere. Moreover the integral function F is Lipschitz continuous on [a,b].
2)B If f is Riemann integrable on [a,b], and G is any Lipschitz continuous function on [a,b] which is differentiable with derivative equal to f at those points where f is continuous, then G does differ from the integral F of f by a constant, i.e. G(x)-G(a) = F(x).
The proof of this version follows from a stronger mean value theorem that shows that a Lipschitz continuous function with derivative zero a.e. is constant. you might try proving this using compactness.
These theorems reach their final form for Lebesgue integrable functions using the still subtler concept of “absolute continuity” in place of Lipschitz continuity.
