# real analysis – Definition of differentiability on closed sets

Let $$Usubset mathbb{R}^{n}$$ be an open set. A function $$f: U to mathbb{R}^{m}$$ is said to be differentiable at $$a in U$$ if there exists some linear map $$Df(a):mathbb{R}^{n}to mathbb{R}^{m}$$ such that:
$$lim_{hto 0}frac{||f(a+h)-f(a)-Df(a)(h)||}{||h||} = 0$$
This is standard. However, during a course, my professor mentioned that we can define differentiability when $$U$$ is not assumed to be open as follows. If $$U$$ is not necessarily open, then $$f$$ is said to be differentiable at $$a in U$$ if there exists some open set $$Vsubset mathbb{R}^{n}$$ and a function $$tilde{f}: V to mathbb{R}^{m}$$ such that $$a in V$$, $$tilde{f}$$ is differentiable at $$a$$ and $$tilde{f}bigg{|}_{Vcap U} equiv f$$.

I understand this definition. But isn’t it possible that there are two different open sets $$V_{1}$$ and $$V_{2}$$ and functions $$tilde{f}_{1},tilde{f}_{2}$$ satisfying the above conditions? And if so, shouldn’t $$tilde{f}_{1}bigg{|}_{V_{1}cap U} = tilde{f}bigg{|}_{V_{2}cap U} equiv f$$? It does not seem to be the case since these functions seem not to have the same domain.