I'm working on this problem and showed that X is contractible. In fact I showed that X has the origin (0,0) as its deformation retract. However, I'm stuck at the second part. It seems intuitively clear that there is no such homopoty, but can't find a contradiction. Could anyone help me?
yes the origin is a strong deformationretract of this one. But here comes the hint: you take the subspace topology (although not stated explicitely, but otherwise there would exist such a basepoint preserving homotopy), therefore you can't "move" all the infinite many points near $x_0$ to $I\times 0$ and tear them apart from $x_0$. This is precisely what we require for a continous function. Try to formalize this situation. (note again: every(!!) neighborhood of $(0,1)$ contains infinitely many points in $X - (0 \times I)$!)
Your proof should also contain that at some point, all points in $(x,I)$ for $x >0$ have to pass through $I \times 0$, which means that we are in the situation above.