There is this problem that goes as follows:
If I have two analytical functions $f,g:I→R$ (where $I$ is an open interval), and I know that there is a point $a∈I$ where $f(a)=g(a)$ and also $f^{(k)}(a)=g^{(k)}(a), \enspace \forall k \in \mathbb{R}$ where $k$ is denoting the degree of the derivative. Then show that$f(x)=g(x) \enspace \forall x\in I$
I know the standard approach to solve this, which is given at this post.
My question is: In the way the questions is presented to me in my book, it is given that we also have $f^{(k)}(a)=g^{(k)}(a)$. But is this simply a consequence to the fact that $f \& g$ are both analytical and equal at the point $a \in I$ ?? If so, haw can I show this?
No, it is not a consequence. Take $f,g\colon\mathbb R\longrightarrow\mathbb R$ defined by $f(x)=0$ and by $g(x)=x$. Then you have $f(0)=g(0)$, but $0=f'(0)\neq g'(0)=1$.