Homomorphism of adic rings
Solution 1:
What you need is $f(I^m) \subset J$ for some $m$.
If there is such a $m$ then $f$ is obviously continuous. No need that $m=1$ as the identity $R=k[[t]],I=(t)\to S=k[[t]],J=(t^2)$ is obviously continuous.
If there is no such $m$ then there is a sequence $a_k\in I^k$ such that $f(a_k)\not\in J$ so that $a_k\to 0$ in $R$ but $f(a_k) \not\to 0$ in $S$, whence $f$ is not continuous.
It is not obvious to me if the condition $f(I_{tnil})\subseteq J_{tnil}$ is sufficient.