Proof That the Identity Function On the Unit Disc is Homotopic to the Constant Function Mapping the Disk to the Origin

Theorem

That the identity function on the unit disk is homotopic to the constant function mapping the disk to (0,0).

Proof

Supposeandare unit disks andis the identity functionfor

Letbe the constant function on the unit diskfor

Letbe a point in the disk.

Letbe point on the line fromtothat is t times the distance fromto

Henceand

The condition for a homotopy is met and the theorem is proved.