]> No Title

8.4. DIFFERENTIABILITY

47

8.3.4. Proposition. If $\mathit{f}\mathrm{,}$ $\mathit{g}\mathrm{\in }{{F}}_{\mathit{a}}$, then

$\mathrm{▵}\mathrm{\left(}\mathit{f}\mathrm{+}\mathit{g}{\mathrm{\right)}}_{\mathit{a}}\mathrm{=}\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{+}\mathrm{▵}{\mathit{g}}_{\mathit{a}}\mathrm{.}$

Proof. Exercise. (Solution .)

The last two propositions prefigure the fact that differentiation is a linear operator; the next result will lead to Leibniz's rule for differentiating products.

8.3.5. Proposition. If $\mathit{\phi }\mathrm{,}$ $\mathit{f}\mathrm{\in }{{F}}_{\mathit{a}}$, then

$\mathrm{▵}\mathrm{\left(}\mathit{\phi }\mathit{f}{\mathrm{\right)}}_{\mathit{a}}\mathrm{=}\mathit{\phi }\mathrm{\left(}\mathit{a}\mathrm{\right)}\mathrm{\cdot }\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{+}\mathrm{▵}{\mathit{\phi }}_{\mathit{a}}\mathrm{\cdot }\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}\mathrm{+}\mathrm{▵}{\mathit{\phi }}_{\mathit{a}}\mathrm{\cdot }\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{.}$

Proof. Problem.

Finally, we present a result which prepares the way for the chain rule.

8.3.6. Proposition. If $\mathit{f}\mathrm{\in }{{F}}_{\mathit{a}}\mathrm{,}$ $\mathit{g}\mathrm{\in }{{F}}_{\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}}$, and $\mathit{g}\mathrm{\circ }\mathit{f}\mathrm{\in }{{F}}_{\mathit{a}}$, then

$\mathrm{▵}\mathrm{\left(}\mathit{g}\mathrm{\circ }\mathit{f}{\mathrm{\right)}}_{\mathit{a}}\mathrm{=}\mathrm{▵}{\mathit{g}}_{\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}}\mathrm{\circ }\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{.}$

Proof. Exercise. (Solution .)

8.3.7. Proposition. Let $\mathit{A}\mathrm{\subseteq }\mathbb{R}\mathrm{.}$ A function $\mathit{f}\mathrm{:}\mathit{A}\mathrm{\to }\mathbb{R}$ is continuous at the point $\mathit{a}$ in $\mathit{A}$ if and only if $\mathrm{▵}{\mathit{f}}_{\mathit{a}}$ is continuous at $\mathrm{0}\mathrm{.}$

Proof. Problem.

8.3.8. Proposition. If $\mathit{f}\mathrm{:}\mathit{U}\mathrm{\to }{\mathit{U}}_{\mathrm{1}}$ is a bijection between subsets of $\mathbb{R}$, then $\mathit{f}\mathit{0}\mathit{o}\mathit{r}$ each $\mathit{a}$ in $\mathit{U}$ the function $\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{:}\mathit{U}\mathrm{-}\mathit{a}\mathrm{\to }{\mathit{U}}_{\mathrm{1}}\mathrm{-}\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}$ is invertible and

$\mathrm{\left(}\mathrm{▵}{\mathit{f}}_{\mathit{a}}{\mathrm{\right)}}^{\mathrm{-}\mathrm{1}}\mathrm{=}\mathrm{▵}\mathrm{\left(}{\mathit{f}}^{\mathrm{-}\mathrm{1}}{\mathrm{\right)}}_{\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}}\mathrm{.}$

Proof. Problem.

8.4. DIFFERENTIABILITY

We now have developed enough machinery to talk sensibly about differentiating real valued functions.

8.4.1. Definition. Let $\mathit{f}\mathrm{\in }{{F}}_{\mathit{a}}$. We say that $\mathit{f}$ is differentiable at $\mathit{a}$ if there exists a linear function which is tangent at $\mathrm{0}$ to $\mathrm{▵}{\mathit{f}}_{\mathit{a}}$. If such a function exists, it is called the differential of $\mathit{f}$ at $\mathit{a}$ and is denoted by $\mathit{d}{\mathit{f}}_{\mathit{a}}$. (Don't be put off by the slightly complicated notation; $\mathit{d}{\mathit{f}}_{\mathit{a}}$ is just a member of $\mathit{L}$ satisfying $\mathit{d}{\mathit{f}}_{\mathit{a}}\mathrm{\simeq }\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{.}$) We denote by ${{D}}_{\mathit{a}}$ the family of all functions in ${{F}}_{\mathit{a}}$ which are differentiable at $\mathit{a}\mathrm{.}$

The next proposition justifies the use of the definite article which modifies ""differential“ in the preceding paragraph.

8.4.2. Proposition. Let $\mathit{f}\mathrm{\in }{{F}}_{\mathit{a}}$. If $\mathit{f}$ is diffe rentiable at $\mathit{a}$, then its diffe rential is unique. (That $\mathit{i}\mathit{s}$, there is at most one linear map tangent at $\mathrm{0}$ to $\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{.}$)

Proof. Proposition . $\mathrm{\square }$

8.4.3. Example. It is instructive to examine the relationship between the differential of $\mathit{f}$ at $\mathit{a}\mathrm{,}$ which we defined in , and the derivative of $\mathit{f}$ at $\mathit{a}$ as defined in beginning calculus. For $\mathit{f}\mathrm{\in }{{F}}_{\mathit{a}}$ to be differentiable at $\mathit{a}$ it is necessary that there be a linear function $\mathit{T}\mathrm{:}\mathbb{R}\mathrm{\to }\mathbb{R}$ which is tangent at $\mathrm{0}$ to $\mathrm{▵}{\mathit{f}}_{\mathit{a}}$. According to there must exist a constant $\mathit{c}$ such that $\mathit{T}\mathit{x}\mathrm{=}\mathit{c}\mathit{x}$ for all $\mathit{x}$ in $\mathbb{R}$. For $\mathit{T}$ to be tangent to $\mathrm{▵}{\mathit{f}}_{\mathit{a}}$, it must be the case that

$\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{-}\mathit{T}\mathrm{\in }\mathrm{0}$;

that is,

$\underset{\mathit{h}\mathrm{\to }\mathrm{0}}{\mathrm{lim}}\frac{\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{\left(}\mathit{h}\mathrm{\right)}\mathrm{-}\mathit{c}\mathit{h}}{\mathit{h}}\mathrm{=}\mathrm{0}\mathrm{.}$