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Much of the elementary theory of differential calculus rests on a few simple properties of the families O and 0. These are given in propositions

8.1.6. Definition. A function L:RR is linear if




for all x, y, cR. The family of all linear functions from R into R will be denoted by L.

The collection of linear functions from R into R is not very impressive, as the next problem shows. When we get to spaces of higher dimension the situation will become more interesting.

8.1.7. Example. A function f:RR is linear if and only if its graph is a (nonvertical) line through the origin.

Proof. Problem.

CAUTION. Since linear functions must pass through the origin, straight lines are not in general graphs of linear functions.

8.1.8. Proposition. Every member of 0 belongs to O; so does every member of L. Every member of O is continuous at 0.

Proof. Obvious from the definitions.

8.1.9. Proposition. Other than the constant function zero, no linear function belongs to 0.

Proof. Exercise. (Solution .)

8.1.10. Proposition. The family O is closed under addition and multiplication by constants.

Proof. Exercise. (Solution .)

8.1.11. Proposition. The fa mily 0 is closed under addition and multiplication by constants.

Proof. Problem.

The next two propositions say that the composite of a function in O with one in 0 (in either order) ends up in 0.

8.1.12. Proposition. If gO and f0, then fog0.

Proof. Problem.

8.1.13. Proposition. If g0 and fO, then fg0.

Proof. Exercise. (Solution .)

8.1.14. Proposition. If φ, fO, then φf0.

Proof. Exercise. (Solution .)

Remark. The preceding facts can be summarized rather concisely. (Notation: C0 is the set of all functions in F0 which are continuous at 0.)

(1) L0OC0.

(2) L0=0.

(3) O+OO; αOO.

(4) 0+00; α00.

(5) 0O0.

(6) O00.

(7) OO0.