Why is there an "implication" rather than and "and" in this definition of the derivative?

Question

I am readig Pugh's Analysis book:

Definition

Let $f:U \to \mathbb{R}^m$ be given where $U$ is an open subset of $\mathbb{R}^n$. The function $f$ is differentiable a $p \in U$ with derivative $(Df)_p = T$ if $T:\mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation and $f(p+v) = f(p)+T(v)+R(v) \implies \lim_{|v| \to 0} \dfrac {R(v)}{|v|}=0$.

Partly due to the missing quantifiers, I'm having trouble understanding why there is a "$\implies$" there rather than a "$\wedge$". Isn't it more natural to say

"T is the derivative if we can write $f(p+v) = f(p)+T(v)+R(v)$ AND $\lim_{|v| \to 0} \dfrac {R(v)}{|v|}=0$"?

I'm having trouble seeing what the impact of changing these would be.

Answer 1

The author means

$$\forall R \left((\forall v ~ f(p+v) = f(p) + T(v) + R(v)) \implies \lim_{|v| \to 0} \frac{R(v)}{|v|} = 0\right)$$

which is taking advantage of how only one function for $R$ satisfies the condition. You are maybe thinking something like

$$ \bigg(R = v \mapsto f(p + v) - f(p) - T(v)\bigg) \land \bigg(\lim_{|v| \to 0} \frac{R(v)}{|v|} = 0\bigg)$$

The problem with the second equation is that it isn't defining $R$ as $v \mapsto f(p + v) - f(p) - T(v)$, it is saying 'if' $R$ is defined as such. Alternatively,

$$\begin{cases} \text{define } R \text{ as } v \mapsto f(p + v) - f(p) - T(v) \\ \lim_{|v| \to 0} \frac{R(v)}{|v|} = 0 \end{cases}$$

is an option, in that case sort of using 'and' in the casual sequential sense, like 'crack the egg and put the egg in the bowl and beat the egg and pour the egg into the pan...'.

Answer 2

I can't speak for Pugh, but your construction

T is the derivative if we can write $f(p+v) = f(p)+T(v)+R(v)$ AND $\lim_{|v| \to 0} \dfrac {R(v)}{|v|}=0$

is awkward because neither the left nor the right side of the "AND" is a property of $T$ that needs checking. The left side is always true; you can always do that. And the right side makes no sense unless $R$ is constrained.

Look at it this way, would you write the following?

T is the derivative if $\lim_{|v| \to 0} \dfrac {R(v)}{|v|}=0$ AND we can write $f(p+v) = f(p)+T(v)+R(v)$