if all continuous map $f: s^1 \to X $ can be extended to $f: D^2 \to X$ then $f: s^1 \to X $ is nullhomotopic

Question

if $f: s^1 \to X $ can be extended to $f: D^2 \to X$ then $f: s^1 \to X $ is nullhomotopic...

i need to proof this, any help or idea?

thanks for your comments.

Answer 1

Hint: consider the cone $(S^1 \times [0,1])/(S^1 \times \{0\}$), and note that since $f$ is defined on the disk, it is also defined on the cone (since they are homeomorphic.) Let $f_t$ be the restriction of $f$ to the cone at height $t$.