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8. DIFFERENTIATION OF REAL VALUED FUNCTIONS
Much of the elementary theory of differential calculus rests on a few simple properties of the families $\mathfrak{O}$ and $0$. These are given in propositions
8.1.6. Definition. A function $L:\mathbb{R}\rightarrow \mathbb{R}$ is linear if
$$
L(x+y)=L(x)+L(y)
$$
and
$$
L(cx)=cL(x)
$$
for all $x, y, c\in \mathbb{R}$. The family of all linear functions from $\mathbb{R}$ into $\mathbb{R}$ will be denoted by $L.$
The collection of linear functions from $\mathbb{R}$ into $\mathbb{R}$ 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 $f:\mathbb{R}\rightarrow \mathbb{R}$ is linear if and only if its graph is $\mathrm{a}$ (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. {\it Every member of} $0$ {\it belongs to} $\mathfrak{O}$; {\it so does every member of L. Every member of} $\mathfrak{O}$ {\it is continuous at} $0.$
Proof. Obvious from the definitions. $\square $
8.1.9. Proposition. {\it Other than the constant function zero, no linear function belongs to} $0.$
Proof. Exercise. (Solution .)
8.1.10. Proposition. {\it The family} $\mathfrak{O}$ {\it is closed under addition and multiplication by constants}.
Proof. Exercise. (Solution .)
8.1.11. Proposition. {\it The fa mily} $0$ {\it is closed under addition and multiplication by constants}.
Proof. Problem.
The next two propositions say that the composite of a function in $\mathfrak{O}$ with one in $0$ (in either order) ends up in $0.$
8.1.12. Proposition. {\it If} $g\in \mathfrak{O}$ {\it and} $f\in 0$, {\it then} $f\mathrm{o}g\in 0.$
Proof. Problem.
8.1.13. Proposition. {\it If} $g\in 0$ {\it and} $f\in \mathfrak{O}$, {\it then} $f\circ g\in 0.$
Proof. Exercise. (Solution .)
8.1.14. Proposition. {\it If} $\phi, f\in \mathfrak{O}$, {\it then} $\phi f\in 0.$
Proof. Exercise. (Solution .)
Remark. The preceding facts can be summarized rather concisely. (Notation: $C_{0}$ is the set of all functions in $\mathcal{F}_{0}$ which are continuous at $0.$)
\begin{center}
(1) $L\cup 0\subseteq \mathfrak{O}\subseteq C_{0}.$
(2) $L\cap 0=0.$
(3) $\mathfrak{O}+\mathfrak{O}\subseteq \mathfrak{O}$; $\alpha \mathfrak{O}\subseteq \mathfrak{O}.$
(4) $0+0\subseteq 0$; $\alpha 0\subseteq 0.$
(5) $0\circ \mathfrak{O}\subseteq 0.$
(6) $\mathfrak{O}\circ 0\subseteq 0.$
(7) $\mathfrak{O}\cdot \mathfrak{O}\subseteq 0.$
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