Local symmetry and smoothness in the space of vector-valued continuous functions

In this article, we characterize the left symmetric points in $C(K,X)$, where $K$ is a compact Hausdorff space and $X$ is a Banach space. We also provide necessary and sufficient conditions for the right symmetric points in $C(K,X)$. Further, we identify the smooth points in the space $C_0(K,X)$, $K$ being locally compact Hausdorff space and $X$ being a Banach space.