I was reading an article about Manifolds.They have defined a $C^{k} $ function in the following way :
Let $M$ and $N$ are two $C^{k}$ manifolds of dimensions $m$ and $n$ respectively.A continuous function $h : M \rightarrow N$ is said to be a $C^{k}$ map if for every point $p \in M$ there is some chart $(U,\varphi)$ at $p$ and some chart $ (V,\psi)$ at $ q = h(p)$ such that $h(U) \subset V$ and $\psi \circ h \circ \varphi^{-1} : \varphi(U) \rightarrow \psi(V)$ is a $C^{k}$ function.
Later the article says that this definition does not depend on choice of charts. How to prove it ? Should I prove that instead of $(U,\varphi)$ and $(V,\psi)$ if I take another charts at $p$ and $q$ respectively the corresponding function must be again $C^{k}$ ?
Also I have another problem with this definition Unless you define what is a continuous function between two manifolds how can we say that $h$ is continuous.We can speak about continuity and even $C^{k}$ functions between subsets of Euclidean spaces.Or is this definition consider $M$ and $N$ as topological spaces only so that $h$ can be a continuous function ? I feel even if we avoid that clause definition for $C^{k}$ map will hold.
You know that for any point $p$, for some choice of charts $(x,U),(y,V)$ around $p$ and $f(p)$ respectively, $yfx^{-1}$ is $C^k$. It follows this happens for every other choice $(z,W),(w,T)$ of charts. Indeed, $wfz^{-1}$ will be smooth iff it is $C^k$ at every point of $W\cap T$. By hypothesis you can cover $W\cap T$ by sets $W\cap T\cap O$ where you can write $wfz^{-1}= (wy^{-1})(yfx^{-1})(xz^{-1})$. But this is a composition of $C^k$ functions, since the first and last terms are transition functions which must be $C^k$, and the middle term is $C^k$ by hypothesis.