It is emphasized in all the materials that I have seen about Hartman-Grobman theorem (including textbooks and Wikipedia) that the topological conjugacy is only a homeomorphism but not diffeomorphism, i.e., it can only be a $C^0$, not $C^1$. So ... why? Is there an example that a $C^0$ homeomorphism is the best thing we can have, and a diffeomorphism can never become a topological conjugacy?
Also, can this map be differentiable or smooth except at the fixed point (i.e. the zero point of the linearized vector field)?