# Homotopy, retraction related question

How to solve the following problem:

$A$ is a strong deformation retract of $X$ such that exists a continuous function $α:X\to I$, $\alpha^{−1}({0})=A$.

(a) If $H:X\times I\to X$ is a homotopy (rel $A$) between $i◦r$ and $1_X$ ($i:A\to X$ is inclusion and $r:X\to A$ is retraction), and if $\phi:X\times I\to I$ is defined with $\phi(x,t):=\min\left(1,\frac{t}{α(x)}\right)$, for $x\notin A$ and $\phi(x,t)=1$, for $x\in A$, prove that the map $D:X\times I\to X$ given with $D(x,t):=H(x,\phi(x,t))$, is continuous.

(b) If $p:E\to B$ is a fibration ($p$ satisfies homotopy lifting property) and if $f:A\to E$ and $g:X\to B$ are continuous mappings (for which diagram is commutative) and $g◦i=p◦f$, prove that exists continuous map $h:X\to E$ so $h◦i=f$ and $p◦h=g$ (two triangles in diagram are commutative).