§ Axioms for definite integration
- Pete Clark 's notes on honors calculus provides a handy axiomatization of what properties the definite integral ought to satisfy.
- 1. If is a constant function, then .
- 2. If for all , then .
- 3. If , then .
§ Proof of fundamental theorem of calculus from the above axiomatization
- Let be any integrable function over . For , we define . Then:
- (a) The function is continuous at every .
- (b) if is continuous at , then is differentiable at and .
- (c) if is continuous and is any antiderivative of , that is, , then .
- First, by coninuity of and compactness of , there exists a such that for all . If , then and thus from axiom 2 and everything holds.
- Thus we assume that . For all , we take .
- By the third axiom, we see that .