Let $(M,d)$ a smooth compact $n$-dimensional manifold such that $n \geq 2$ supported by metric $d$. Take some $\varepsilon > 0$ and consider $\{(x_{i},y_{i}): i \in \{1,...,k\}$ and $d(x_{i},y_{i}) \leq \varepsilon \}$ and $x_{i} \neq x_{j}$ for all i, j $\in \{1,...,k\}$, as are the $y_{i}'s$. Then there exist a diffeomorphism $g$ of $M$ such that $g(x_{i}) = y_{i}$ and $g$ is $7\varepsilon - C^{0}$ close to the identity.
I'd appreciate any help.
Hint : construct a vector field which send $x_i$ to $y_i$ and which is zero outside a little neighborhood of a path connecting each $x_i$ and $y_i$, using bump functions. The flow of this vector field will gives you the desired diffeomorphism.