515 – The fundamental theorem of calculus

Suppose that f\in{\mathcal R}[a,b] and f has an antiderivative G. Then

\displaystyle G(b)-G(a)=\int_a^b f(t)dt.

Note we are assuming f is Riemann integrable. This means that given \epsilon>0 we can find an \eta>0 such that if P is a tagged partition of {}[a,b] and \Delta(P)<\eta, then

\displaystyle \left|R(f,P)-\int_a^b f(t)dt\right|<\epsilon.

Recall that a tagged partition P consists of a partition \Phi of {}[a,b], represented by a finite sequence of points

a=x_0<x_1<\dots<x_n=b,

and a sequence of representatives of the intervals defined by this partition, i.e., a collection of points x_i^*\in[x_{i-1},x_i] for i=1,\dots,n.

We denote by \delta x_i the number x_i-x_{i-1} and by \Delta(P)=\Delta(\Phi) the norm of P,

\displaystyle \Delta(P)=\max_{i=1,\dots,n}\delta x_i.

The Riemann sum associated to f and P is the sum

\displaystyle R(f,P)=\sum_{i=1}^n f(x_i^*)\delta x_i.

Pick any partition \Phi=\{a=x_0<x_1<\dots<x_n=b\} of {}[a,b] with \Delta(\Phi)<\eta. We want to define a particular tagged partition P with underlying partition \Phi by appealing to the fact that f has an antiderivative G. Specifically, by the mean value theorem, we have that for all i=1,\dots,n, there is some x_i^*\in[x_{i-1},x_i] such that

G(x_i)-G(x_{i-1})=f(x_i^*)\delta x_i.

Define P in terms of the points x_i^* and the partition \Phi. Then

R(f,P)=\sum_{i=1}^n f(x_i^*)\delta x_i=\sum_{i=1}^n (G(x_i)-G(x_{i-1})) \displaystyle =G(b)-G(a).

By our choice of \eta, we know that |R(f,P)-\int_a^b f(t)dt|<\epsilon, so

\displaystyle \left|G(b)-G(a)-\int_a^b f(t)dt\right|<\epsilon.

But the left hand side is independent of \epsilon, and \epsilon was arbitrary. It follows that \int_a^b f(t)dt=G(b)-G(a), as we wanted.

The same argument, but restricting ourselves to the interval {}[a,x], shows that

\displaystyle G(x)-G(a)=\int_a^x f(t)dt = F(x) -F(a),

where F(x)=\int_a^x f(t)dt for x\in[a,b], so in particular F(a)=0. It follows that G and F differ by a constant, and therefore, if f is Riemann integrable and has an antiderivative at all, then F is such an antiderivative.

The question remains of what Riemann integrable functions f (with the intermediate value property) have antiderivatives. Another natural question has to do with the fact that our definition of antiderivative is very restrictive; it also makes sense to simply ask whether the equality F'(x)=f(x) must hold for some x, assuming only that f is integrable. It turns out that both questions require the introduction of the Lebesgue integral to be answered in a satisfactory way.

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