Question about homotopy form Hatcher's Algebraic Topology.

Question

In Algebraic topology by Hatcher, there is a statement saying that
A homotopy $f_t:X\rightarrow X$ that gives a deformation retraction of $X$ onto a subspace $A$ has the property that $f_t(A)=identity$ for all $t$.
I don't understand why this happens. In some steps of deformation retraction the functions $f_t$ many not be identical on $A$, we only look in the initial and final steps. I have a big confusion about it.
Please help me to understand that line.

Answer 1

This terminology varies some across authors. Often an author will define a deformation retraction of $X$ onto $A \subset X$ to be a continuous map $X \times [0, 1] \to X$ satisfying (for all $x \in X$, $a \in A$) $$H(x, 0) = x, \qquad H(x, 1) \in A, \qquad H(a, 1) = a$$ and define a strong deformation retraction to be a deformation retraction that additionally satisfies $$H(a, t) = a$$ for all $a \in A$, $t \in [0, 1]$ (obviously this condition implies the condition that $H(a, 1) = a$ for all $a \in A$). But some authors, including Hatcher (see $\S$ 0, page 2), define a deformation retraction to satisfy this last condition, too.