When I learn "An introduction to Sobolev space and interpolation space" by Luc Tartar, Chapter 22"Real interpolation; K-method", I am confused by the continuity of $K(t;a)$, Let $E_{0}$ and $E_{1}$ be two normed spaces, continuously embedded into a topological vector space $E$ so that we can define $K(t;a)=\inf_{a=a_{0}+a_{1}}\big(\|a_{0}\|_{0}+t\|a_{1}\|_{1}\big)$.
Then we have
(1) $K(t;a)$ is nondecreasing in $t$
(2) $\frac{K(t;a)}{t}$ is nonincreasing in $t$
(3) $K(t;a)$ is concave in $t$, as an infimum of affine functions, so that it is continuous. Why? Can "infimum of affine functions" infers that "it is continuous"?
