Extension continuous function not onto then restriction is nullhomotopic Let $F:\overline{D^{n+1}}\to \overline{D^{n+1}}$ continuous function such that $F(S^n)\subset S^n$. ($\overline{D^{n+1}}$ closed disc radio 1)
Let $f:S^n\to S^n$ by $f(x)=F(x)$.
Show that $F$ onto or $f$ is nullhomotopic.
The above is equivalent to proving that if $F$ is not onto, then $f$ nullhomotopic.
On the previous proposal, there is a very similar one, but instead of $F$ not onto, we have that  $f$ not onto implies that $f$ is nullhomotopically.
The proof is as follows:
If $f$ is not onto, exists $p\in S^n$ such that $p\not\in f(S^n)$. Let $h:S^n-\left\{p\right\}$ homeomorphic to $\mathbb{R}^{n}$ (stereographic proyection)
Now $h\circ f:S^n\to \mathbb{R}^n$ is continuous  and $\mathbb{R}^n$ contractible, so $h\circ f$ is nullhomotopic. Since $h\circ f$ is nullhomotopic, then $h^{-1}\circ (h\circ f)=f$ is nullhomotopic, i.e.  $f$ nullhomotopic.
How to use the hypothesis that $F$ is not onto in this case?
 A: Since $F$ is not onto, we find a point $x \in \overline{D^{n+1}}$ such that $x$ is not in the image of $F$. Moreover, $F(\overline{D^{n+1}})$ is compact, since images of compact sets under continuous functions are compact. Hence, we find $\epsilon > 0$ such that $B_\epsilon(x) \cap \overline{D^{n+1}}$ and the image of $F$ are disjoint (indeed, compact subsets of $\overline{D^{n+1}}$ are precisely those which are closed in the topology of $\mathbb R^{n+1}$).
If now $x \in S^n$, then $f$ is clearly nullhomotopic; you gave that argument in your question. Suppose for a contradiction that $f$ was onto. Then $x \notin S^n$, so the image of $F$ is not contractible*, a contradiction. Thus, $x \in S^n$ and we are reduced to the case where $f$ is not onto.
*Note: $\overline{D^{n+1}} \setminus B_\epsilon(x)$ deformation retracts to $S^n$, and either the image of $F$ composed with the retraction is not all of $S^n$ and hence contractible (so $F$ and hence $f$ is nullhomotopic), or the image of the composition is not contractible and in fact equals $S^n$. But this is not possible since it's the image of a contractible space under a continuous function.
