can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Question

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then

My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that $\|h\|_{L^{p}(\mathbb R)} < \epsilon,$ and $$h(x)=f(x)-f(\gamma_{0});$$ for all $x$ in some neighbourhood of $\gamma_{0}$ ? (Bit roughly speaking, question states that, can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point $\gamma_{0}.$)

Thanks,

Answer 1

Consider $h_n = f \cdot \chi_{\Bbb{R}^n \setminus B_{1/n}(\gamma_0)}$ and use dominated convergence.

This implies that $h_n \to f$ in $L^p$ and each $h_n$ is constant on the ball $B_{1/n}(\gamma_0)$ with radius $1/n$ around $\gamma_0$.

EDIT: Ok, to get a solution to the precise formulation of your problem, take

$$ h(x) = (f(x) - f(\gamma_0)) \cdot \chi_{B_{1/n}(\gamma_0)}. $$

You can then conclude (using dominated convergence and the fact that the measure of the ball decreases to zero) that $h \to 0$ in $L^p$ as $n \to \infty$.