I have a midterm tomorrow and stumbled upon this problem: $M$ is a compact $m$-manifold and $f:M\times(-1,1) \to M$ is a smooth map such that $f|_{M\times\{0\}}$ is a diffeomorphism. Prove that for some $\epsilon>0$, $f|_{M\times\{t\}}$ is a diffeomorphism for all $|t|<\epsilon.$
I'm not sure how to proceed, but here's my attempt: For $p\in M$ take a chart centered at $p$ $(U,\phi)=(U,x_1,x_2,\dots,x_m)$. Since $f|_{M\times\{0\}}$ is a diffeomorphism, the jacobian $(\frac{\partial f_i}{\partial x_j})_{1\leq i\leq m, 1\leq j\leq m+1}$ at $0$ is invertible, $(x_{m+1}=t\in(-1,1))$, so there exists neighborhood $U_p\times (-\epsilon_p,\epsilon_p)$ such that the restriction of $f$ on it is a diffeomorphism. Now since $M$ is compact, then we can choose finitely many $U_p$ to cover it, so let $M=\cup_iU_{p_i}.$ Then by choosing a partition of unity $\{\rho_i\}$ subordinate to $\{U_{p_i}\}$, we can express $f=\sum\rho_if|_{U_{p_i}\times \{t\}}$, where $|t|<\epsilon_{p_i}~\forall i$. By letting Let $\epsilon=\min_i(\epsilon_{p_i})$, we arrive at the conclusion.