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I was prompted by some recent readings, and also by this question, to try to rectify the fact that my notions of discrete and continuous differentiation have slipped away from one another.

I'm most shaky on the discrete side. I think a discrete (directional) derivative can be defined by $\frac{f(x+h)-f(x)}{||h||}$, which as best I can tell, is a quantity that is most comfortable when $f:A^n\to A^1$ is a function between affine spaces. (I am assuming the codomain has dimension 1 only for simplicity; presumably it is easy to extend this to a total derivative.) In that case we have that $\Delta f:A^n\to\Bbb R$

However, for the continuous derivative, I know that the derivative is most comfortable when $f:M^n\to M^1$ is a function on a manifold, and in that case $Df$ is a member of the tangent bundle $T_\bullet M^n$. I'm not terribly sure how to directionalize it to make the analogy exact, but maybe an inner product trick would work?

My question is: Do I have the correct understanding of a discrete derivative, and if so, why is an affine space the discrete analogue of a tangent bundle?

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I think it would be helpful to review the definition of the total or Frechet derivative, $$\lim_{h \to 0} \frac{ \| f(x + h) - f(x) - Df_x(h) \| }{ \|h\| } = 0$$ It is defined for maps between Banach (complete normed vector) spaces $f:V\to W$ and if exists it serves as linearisation of $f$ at the point $x$, $Df:V\to W$, $Df\in \mathcal L(V,W)$. For finite dimensional spaces, in given bases, it is represented by Jacobian matrix (at each $x$).

In differential geometry the total derivative (pushforward) is defined for smooth maps between manifolds $\varphi:M\to N$,and is found as a Frechet derivate in chosen coordinate charts.

In the special case $f:M\to \mathbb R$, $df$ is a member of cotangent bundle: at each $x\in M$ it's an element of the dual space, a 1-form, linear functional on the tangent space, such that when given a tangent it returns the rate of change of the function $f$ in that "direction".

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