cone of a $C^*$-algebra

Question

Let $A$ be a $ C^*$-algebra.The cone of $A$ is defined to be $CA:=\{f\in C([0,1]):f(0)=0\}$,the suspension of $A$ is defined to be $SA:=\{f\in C([0,1]): f(0)=f(1)=0\}$,then $0\longrightarrow SA\overset{\iota}{\longrightarrow} CA\overset{\psi}{\longrightarrow} A\longrightarrow 0.$ is an exact sequence.

My question is: why the map$CA\overset{\psi}{\longrightarrow} A$ is surjective? For any $a\in A$,how to show that there must exist a map $f\in CA$ such that $f(1)=a$?

Answer 1

Simply take the mapping given by $f(t) = ta$.