This question I found in Elementary Classical Analysis Exercise 6.1.3
Let $ A = \{(x,y) \in \mathbb{R}^2 | x \in [0,1], y =0 \}$. Suppose $f: A \rightarrow \mathbb{R}^m $ is differentiable at $x_0$. Prove $Df(x_0)$ is not uniquely determined by $f$.
I was able to get $T_1(x) = T_2(x), \forall x \in A$. However I feel this is not enough for me to state that $\exists x \in \mathbb{R}^2 | T_1(x) \neq T_2(x)$. How can I make my argument more convincing, because intuitively I know that this has to be not equal since we can only approach $x_0$ on the interval given. My alternative way is to give an example of two linear transformations that would be legit only within $A$, but honestly I don't know how to construct the example using the right language.
Thanks!