Definition. Let $M$ and $N$ be smooth manifolds, and let $F\colon M\to N$ be any map. We say that $F$ is a smooth map if for every $p\in M$, there exists smooth chart $\big(U,\varphi\big)$ containing $p$ and $\big(V,\psi\big)$ containing $F(p)$ such that $F(U)\subseteq V$ and the composite map $\psi\circ F\circ \varphi^{-1}\colon \varphi(U)\to\psi(V)$ is smooth.
I have to show the following proposition, but many points are not clear to me.
Proposition. Suppose $M$ and $N$ are smooth manifolds with or without boundary, and $F\colon M\to N$ is a map. Then $F$ is smooth if and only if either of the following condition is satisfied:
$(a)$ For every $p\in M$, there exist smooth charts $(U,\varphi)$ containing $p$ and $(V,\psi)$ containing $F(p)$ such that $U\cap F^{-1}(V)$ is open in $M$ and the composite map $\psi\circ F\circ\varphi^{-1}$ is smooth from $\varphi(U\cap F^{-1}(V))$ to $\psi\big(V\big)$.
$(b)$ $F$ is continuous and there exists amooth atlases $\{(U_\alpha,\varphi_\alpha)\}$ and $\{(V_\beta,\psi_\beta)\}$ for $M$ and $N$ respectevely such that for each $\alpha$ and $\beta$, $\psi_\beta\circ F\circ \varphi_\alpha^{-1}$ is a smooth map from $\varphi_\alpha(U_\alpha\cap F^{-1}(V_\beta))$ to $\psi(V_\beta)$.
Proof. I would like to proceed, hoping that it is the simplest way as follows: first prove that $F$ is smooth iff $(a)$ holds, and after showing that $(a)$ holds iff $(b)$ holds.
F is smooth iff $(a)$ holds
($\Rightarrow$) If $F$ is smooth we have that $F(U)\subseteq V$, then $U\subseteq F^{-1}(V)$. Therefore $U\cap F^{-1}(V)=U$ that is open.
$(\Rightarrow)$ If $(a)$ holds for every $p\in M$, there exist smooth charts $(U,\varphi)$ containing $p$ and $(V,\psi)$ containing $F(p)$ such that $U\cap F^{-1}(V)$ is open in $M$ and the composite map $\psi\circ F\circ\varphi^{-1}$ is smooth from $\varphi(U\cap F^{-1}(V))$ to $\psi\big(V\big)$. It only remains to show that $F(U)\subseteq V$, now if $\tilde{p}\in U$ then there exists a smooth chart $(\tilde{V},\tilde{\psi})$ containing $F(\tilde{p})$
Question 1. How can I prove that $F\big(\tilde{q}\big)\in V$?
(a)$\iff$(b)
$(b)\Rightarrow (a)$ it seems obvious.
Question 2. I have no idea how to show that $(a)\Rightarrow (b)$. Could you give me a hints?
Thanks!