Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology.

The evaluation map $$ev\colon N\times\mathcal{C}^\infty(N,X)\rightarrow X$$ is continuous. (This can e.g. be found in "Margalef-Roig/Dominguez - Differential Topology", Proposition 9.6.7)

Therefore one gets a map of sets $$\mathcal{C}(M,\mathcal{C}^\infty(N,X))\rightarrow \mathcal{C}(M\times N,X).$$

I searched the literature for a reference to the "other direction", but did not find anything. I.e. starting with a smooth map $M\times N\rightarrow X$ I want to know whether the adjoint $M\rightarrow \mathcal{C}^\infty(N,X)$ is continuous or if not, in which cases.

Differential topology, Exr.2.4.2 (a starred exercise) is essentially identical to your question, without suggesting an answer: "Under what conditions is the natural map $C_S(X,C_S(Y,Z)) \to C_S(X\times Y, Z)$ a homeomorphism?" $\endgroup$ – Igor Khavkine Jun 15 '15 at 12:38isthe Whitney $C^0$ topology, and Peter Michor's argument works in the continuous case as well. $\endgroup$ – Pedro Lauridsen Ribeiro Jun 17 '15 at 5:02