Let $E_i$ be a $\mathbb R$-Banach space, $\Omega_i\subseteq E_i$, $x_1\in\Omega_1$ and $f:\Omega_1\to\Omega_2$ be $C^1$-differentiable at $x_1$ (see below).
Question 1: Can we show that $f$ is continuous at $x_1$?
Question 2: If $\gamma$ is a $C^1$-curve on $\Omega_1$ through $x_1$ (see below), can we show that $(f\circ\gamma)'(0)\in T_{f(x)}\:\Omega_2$?
Regarding question 1: Let $d_1$ denote the metric induced by the norm on $E_1$, $\varepsilon>0$ and $(O_1,\tilde f)$ be a $C^1$-extension of $f$ at $x_1$. Since $\tilde f$ is continuous at $x_1$, there is a $\delta>0$ with $$\left\|\tilde f(x_1)-\tilde f(y_1)\right\|_{E_2}<\varepsilon\;\;\;\text{for all }y_1\in O_1\text{ with }d_1(x_1,y_1)<\delta\tag1$$ and hence $$\left\|f(x_1)-f(y_1)\right\|_{E_2}<\varepsilon\;\;\;\text{for all }y_1\in O_1\cap\Omega_1\text{ with }d_1(x_1,y_1)<\delta\tag2.$$ This should yield the claim, since $\{y_1\in O_1\cap\Omega_1:d_1(x_1,y_1)<\delta\}$ is an $\Omega_1$-open neighborhood of $x_1$. Or am I missing something?
Regarding question 2: It's trivial to see that $f\circ\gamma$ is $C^1$-differentiable at $0$, $(f\circ\gamma)(0)=f(x)$ and $(f\circ\gamma)(I)\subseteq\Omega_2$. So, the claim is obviously true. But can we somehow determine the smallest (arbitrary/closed/open) set $\tilde\Omega_2\subseteq\Omega_2$ usch that $(f\circ\gamma)'(0)\in T_{f(x)}\:\tilde\Omega_2$?
. But in order to conclude that $f\circ\gamma$ is a $C^1$-curve on $\Omega_2$ through $f(x)$, we need to show that there is a nontrivial subset $\tilde I$ of $I$ with $0\in\tilde I$ and $
Definition 1: $f$ is called $C^1$-differentiable at $x_1$ if $$\left.f\right|_{O_1\:\cap\:\Omega_1}=\left.\tilde f\right|_{O_1\:\cap\:\Omega_1}\tag1$$ fomr some $\tilde f\in C^1(O_1,E_2)$ for some $E_1$-open neighborhood $O_1$ of $x_1$. $(O_1,\tilde f)$ is called $C^1$-extension of $f$ at $x_1$.
Definition 2: Let $M$ be a subset of a $\mathbb R$-Banach space and $x\in M$. $(I,\gamma)$ is called $C^1$-curve on $M$ through $x$ if $I\subseteq\mathbb R$ is a nontrivial interval with $0\in I$ and $\gamma:I\to M$ is $C^1$-differentiable with $\gamma(0)=x$. Let $$T_x\:M:=\{\gamma'(0):\gamma\text{ is a }C^1\text{-curve on }M\text{ through }x\}.$$