I would like of a proof to the following fact: if $f : M \to \mathbb{R}$ is a continuous function that attains the values $a < b$, then for any $c\in (a,b)$ there is diffeomorphism $\varphi : M\to M$ such that $f \circ \varphi$ is close to $c$ in the $L^p(M)$ norm for every $1\leq p < \infty$.

Let $M$ be a compact and connected manifold without boundary. My question is how to prove the following fact which I believe is true:

If $f : M \to \mathbb{R}$ is a continuous function that attains the values $a < b$, then for any $c\in [a,b]$ and any $1\leq p<\infty$, there is diffeomorphism $\varphi : M\to M$ such that $f \circ \varphi$ is close to $c$ in the $L^p(M)$ norm.

It seems that it is a classical result that if $f : M \to \mathbb{R}$ is a continuous function negative somewhere, but not constant, where $M$ is a compact connected manifold, then there is a smooth diffeomorphism $\varphi$ of $M$ such that $f\circ \varphi$ is close of the constant function $-1$ on $M$ on the $L^p(M)$ norm, for every $p\in [1,\infty)$. This is used for instance on the solution of the Kazdan-Warner problem, without any proof.

It was later pointed out that if $f$ assume the values $a < b$, then for any $c\in (a,b)$ there is such $\varphi$ such that $f \circ \varphi$ is close to $c$. How one proves this?

I would like of a proof to the following fact: if $f : M \to \mathbb{R}$ is a continuous function that attains the values $a < b$, then for any $c\in (a,b)$ there is diffeomorphism $\varphi : M\to M$ such that $f \circ \varphi$ is close to $c$ in the $L^p(M)$ norm for every $1\leq p < \infty$.

It seems that it is a classical result that if $f : M \to \mathbb{R}$ is a continuous function negative somewhere, where $M$ is a compact connected manifold, then there is a smooth diffeomorphism $\varphi$ of $M$ such that $f\circ \varphi$ is close of the constant function $-1$ on $M$ on the $L^p(M)$ norm, for every $p\in [1,\infty)$. This is used for instance on the solution of the Kazdan-Warner problem, without any proof.

How the proof goes?

It seems that it is a classical result that if $f : M \to \mathbb{R}$ is a continuous function negative somewhere, but not constant, where $M$ is a compact connected manifold, then there is a smooth diffeomorphism $\varphi$ of $M$ such that $f\circ \varphi$ is close of the constant function $-1$ on $M$ on the $L^p(M)$ norm, for every $p\in [1,\infty)$. This is used for instance on the solution of the Kazdan-Warner problem, without any proof.

It was later pointed out that if $f$ assume the values $a < b$, then for any $c\in (a,b)$ there is such $\varphi$ such that $f \circ \varphi$ is close to $c$. How one proves this?