Differential in a point. Is that a function?

Question

I was reading more on differentials and I found the phrasing "let there be a point $c$ in a neighborhood $D$ (...) $Df(c)$ is a one to one function". I understand that for a differential we also have a direction, but I fail to understand how a differential in a fixed point can be a function. I might have understanding problems with the definition of a differential and I would appreciate it if someone could explain it to me better. Thank you.

Answer 1

The differential of a function $f$ at a point $c$ is the best linear approximation $L$ of the values of the function increments in a neighbourhood of $c$, in a very precise sense: $$f(c+h)-f(c)=L\cdot h+o(h).$$

Answer 2

Given a differentiable function $f:\Bbb R^m\to \Bbb R^n$ and $c\in \Bbb R^m$ a point. Then the differential of $f$ at $c$ is a linear map $Df(c):\Bbb R^m\to\Bbb R^n$ given by $$ f(c+x)\approx f(c)+Df(c)(x) $$ for a certain rigorously defined meaning of $\approx$.

As an example, if $f:\Bbb R\to\Bbb R$ is the function $f(x)=x^2$, you might be used to thinking of its derivative at $c$ as the number $2c$. In this framework, the differential of $f$ at $c$ is the linear function $x\mapsto 2c\cdot x$. In general, the differential is represented in matrix form by the Jacobian matrix.