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How does one prove continuity with $f$ restricted into a set?

Let $f: (M_1,d_1)\to (M_2,d_2)$ be continuous and $f(M_1)\subseteq N_2\subseteq M_2$ then $f: (M_1,d_1)\to(N_2,d_2)$ is continuous.

I need a hint for this one.

Let $(M_1,d_1),\ldots,(M_n,d_n),$ and $M=M_1\times\cdots\times M_n.$ Prove the following:

$f: (N,d)\to (M,d_e)$ is continuous iff $f_i: (N,d)\to(M_i,d_i)$ is continuous for $1\le i\le n.$

The left implication is easy because if each $f_i$ is continuous then clearly $f$ is continuous, but don't know how to prove the right implication.

Thanks!