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8.4. DIFFERENTIABILITY
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8.3.4. Proposition. {\it If} $f, g\in \mathcal{F}_{a}$, {\it then}
$$
\triangle(f+g)_{a}=\triangle f_{a}+\triangle g_{a}.
$$
Proof. Exercise. (Solution .)
The last two propositions prefigure the fact that differentiation is a linear operator; the next result will lead to {\it Leibniz}'{\it s rule} for differentiating products.
8.3.5. Proposition. {\it If} $\phi, f\in \mathcal{F}_{a}$, {\it then}
$$
\triangle(\phi f)_{a}=\phi(a)\cdot\triangle f_{a}+\triangle\phi_{a}\cdot f(a)+\triangle\phi_{a}\cdot\triangle f_{a}.
$$
Proof. Problem.
Finally, we present a result which prepares the way for the {\it chain rule}.
8.3.6. Proposition. {\it If} $f\in \mathcal{F}_{a}, g\in \mathcal{F}_{f(a)}$, {\it and} $g\circ f\in \mathcal{F}_{a}$, {\it then}
$$
\triangle(g\circ f)_{a}=\triangle g_{f(a)}\circ\triangle f_{a}.
$$
Proof. Exercise. (Solution .)
8.3.7. Proposition. {\it Let} $A\subseteq \mathbb{R}.$ {\it A function} $f:A\rightarrow \mathbb{R}$ {\it is continuous at the point} $a$ {\it in} $A$ {\it if and only if} $\triangle f_{a}$ {\it is continuous at} $0.$
Proof. Problem.
8.3.8. Proposition. {\it If} $f:U\rightarrow U_{1}$ {\it is a bijection between subsets of} $\mathbb{R}$, {\it then} $f0or$ {\it each} $a$ {\it in} $U$ {\it the function} $\triangle f_{a}:U-a\rightarrow U_{1}-f(a)$ {\it is invertible and}
$$
(\triangle f_{a})^{-1}=\triangle(f^{-1})_{f(a)}.
$$
Proof. Problem.
8.4. DIFFERENTIABILITY
We now have developed enough machinery to talk sensibly about {\it differentiating} real valued functions.
8.4.1. Definition. Let $f\in \mathcal{F}_{a}$. We say that $f$ is differentiable at $a$ if there exists a linear function which is tangent at $0$ to $\triangle f_{a}$. If such a function exists, it is called the differential of $f$ at $a$ and is denoted by $df_{a}$. (Don't be put off by the slightly complicated notation; $df_{a}$ is just a member of $L$ satisfying $df_{a}\simeq\triangle f_{a}.$) We denote by $\mathcal{D}_{a}$ the family of all functions in $\mathcal{F}_{a}$ which are differentiable at $a.$
The next proposition justifies the use of the definite article which modifies ``differential'' in the preceding paragraph.
8.4.2. Proposition. {\it Let} $f\in \mathcal{F}_{a}$. {\it If} $f$ {\it is diffe rentiable at} $a$, {\it then its diffe rential is unique}. ({\it That} $is$, {\it there is at most one linear map tangent at} $0$ {\it to} $\triangle f_{a}.$)
Proof. Proposition . $\square $
8.4.3. Example. It is instructive to examine the relationship between the differential of $f$ at $a,$ which we defined in , and the derivative of $f$ at $a$ as defined in beginning calculus. For $f\in \mathcal{F}_{a}$ to be differentiable at $a$ it is necessary that there be a linear function $T:\mathbb{R}\rightarrow \mathbb{R}$ which is tangent at $0$ to $\triangle f_{a}$. According to there must exist a constant $c$ such that $Tx=cx$ for all $x$ in $\mathbb{R}$. For $T$ to be tangent to $\triangle f_{a}$, it must be the case that
$$
\triangle f_{a}-T\in 0;
$$
that is,
$$
\lim_{h\rightarrow 0}\frac{\triangle f_{a}(h)-ch}{h}=0.
$$
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