Reference request: For manifolds $M,N$ the space $C^0(M,N)$ is in general not path-connected. I am currently looking at a paper analysing the path-connectedness of Sobolev spaces between manifolds. One theorem states that under certain conditions the Sobolev space is path-connected iff the space of continuous functions between manifolds $C^0(M,N)$ is path-connected. It is remarked that this is, in general, is not the case (, which in turn motivates further procedure). 
Does anyone know a reference for the remark?

In general $C^0(M, N)$ is not path connected. If it is path connected, it means that all $f:M\to N$ are homotopic to a constant map. There are lots of examples where this is not true: 


*

*If $\pi_1(N)$ is nontrivial, $C^0(\mathbb S^1, N)$ is not path connected (similarly on higher homotopy groups),

*Then $M = N$ are compact, $C^0(N,N)$ is not path connected as the identity map $I : N\to N$ induces identity map on top homology, while constant map induces zero map on top homology, 
