Q1: What is not a correct definition for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ being differentiable at $x_0$?

A1: $\displaystyle \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x - x_0}$ exists.

A2: There is a function $\Delta_{f,x_0} : \mathbb{R} \rightarrow \mathbb{R}$ that is continuous at $x_0$ and satifies $f(x) = f(x_0) + (x - x_0) \cdot \Delta_{f,x_0}(x)$ for all $x \in \mathbb{R}$.

A3: There is a number $b \in \mathbb{R}$ and a function $r : \mathbb{R} \rightarrow \mathbb{R}$ that is continuous at $x_0$ with $r(x_0) = 0$ and satifies $f(x) = f(x_0) + (x - x_0) \cdot b + (x - x_0) \cdot r(x)$ for all $x \in \mathbb{R}$.

A4: $f$ is continuous at $x_0$ and the limit $\displaystyle \lim_{x \rightarrow x_0} \frac{f(x-x_0) - f(x_0)}{x - x_0}$ exists.

Q2: What is the correct definition for a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ being totally differentiable at $\tilde{x}$?

A1: There is a linear map $\ell: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and a function $\phi: \mathbb{R}^n \rightarrow \mathbb{R}^m$ with $$ f(\tilde{x} + h) = f(\tilde{x}) + \ell(h) + \phi(h) $$ and $\frac{\phi(h)}{| h|} \xrightarrow[]{h \rightarrow 0} 0$.

A2: There is a linear map $\ell: \mathbb{R}^n \rightarrow \mathbb{R}^m$ with $$ f(\tilde{x} + h) = f(\tilde{x}) + \ell(h) $$ and $\frac{\ell(h)}{| h|} \xrightarrow[]{h \rightarrow 0} 0$.

A3: There is a linear map $\ell: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and a function $\phi: \mathbb{R}^n \rightarrow \mathbb{R}^m$ with $$ f(\tilde{x} + h) = f(\tilde{x}) + \ell(h) + \phi(h) $$ and $\frac{\ell(h)}{| h|} \xrightarrow[]{h \rightarrow 0} 0$.

A4: There is a linear map $\ell: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and a function $\phi: \mathbb{R}^n \rightarrow \mathbb{R}^m$ with $$ f(\tilde{x} + h) = f(\tilde{x}) + \ell(h) + \phi(h) $$ and $\frac{\ell(h)}{| h|} \xrightarrow[]{h \rightarrow 0} 1$.

Q3: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $$ f(x_1, x_2) = x_1 + x_2 $$ Is the function everywhere totally differentiable.

A1: Yes!

A2: No, not at all points.

A3: No, at no point at all!

A4: One needs more information.