Weak${}^{\ast }$-weak points of continuity and very smooth points in Bochner function spaces $L^p(\mu ,X)$

Suppose that $([0,1],\mathcal{A},\mu )$ is the Lebesgue measure space and $1<p<\mathrm{\infty }$ with $\frac{1}{p}+\frac{1}{q}=1$. We prove that an unit vector $h\in L^p(\mu ,X{)}^{\ast }$ is a weak${}^{\ast }$-weak point of continuity of $S(L^p(\mu ,X{)}^{\ast })$ if and only if $\frac{h(t)}{\mathrm{\parallel }h(t)\mathrm{\parallel }}$ is a weak${}^{\ast }$-weak point of continuity of $S(X^{\ast })$ for almost every $t\in [0,1]$ with $h(t)\mathrm{\ne }0$, where $X$ is a Banach space such that $X^{\ast }$ is separable. As an application of this result, under the same hypothesis on $X$, we prove that a point $f\in L^p(\mu ,X{)}_1$ is a very smooth point if and only if $\frac{f(t)}{\mathrm{\parallel }f(t)\mathrm{\parallel }}\in S(X)$ is a very smooth point for almost every $t\in [0,1]$ with $f(t)\mathrm{\ne }0$. We give several examples to show that these results does not hold for $p=1$.