Let $f: \mathbb{R}^d \mapsto \mathbb{R}$ be $p$-times continuously differentiable with derivatives bounded in $L_\infty$ norm by a constant $M$. Let $B_d(x, r)$ be the ball of radius $r > 0$ centered at $x \in \mathbb{R}^d$, and let $V(B_d(x, r))$ be its volume (Lebesgue measure). Finally let $X$ be a bounded subset of $\mathbb{R}^d$. Can anyone show or disprove that there exists a function $K(x)$ such that, for all $x \in X$:
$$\biggr|\frac{1}{V(B_d(x, r))}\int_{B_d(x, r)} f(z) dz - f(x)\biggr| \leq K(x)r^p\;?$$
Moreover, if the above were true, would it be possible to show that there exists a constant $K$ that satisfies the above for all $x \in X$? (Presumably yes, given that $X$ is bounded)
For context, I'm trying to link the smoothness condition above, which is often used in the literature on KNN-regression, to more standard notions of smoothness such as bounded derivatives. Would appreciate any intuition, links, or general knowledge, thank you!