[Math] Every map $S^1 → X$ is homotopic to a constant map$\implies$Every map $S^1 → X$ extends to a map $D^2 → X$.

Question

I am working at this problem:

Prove:Every map $S^1 → X$ is homotopic to a constant map$\implies$Every map $S^1 → X$ extends to a map $D^2 → X$.

And I am attempting to understand this solution:

It seems to me that this proof only use the fact that $f_t$ is a homotopy and thus continous. But I cannot see that where do we use the condition that $f$ is homotopic to a constant map. So where does this condition apply? Do we necessarily need it?

Answer 1

This fact is used to show that $g(0)=g(0e^{i\theta})=f_0(e^{i\theta})$ is well-defined and does not depend on $\theta$.