Map from 2-sphere into $(\mathbb R^3, |\cdot|)$ Can you help me with this?

Let $S^2 := \{x\in \mathbb R^3:||x||_2 = 1\} \subset (\mathbb R^3, ||\cdot||_2)$  and $T:S^2 \to (\mathbb R, |\cdot|)$ be a continuous map. Since $S^2$ is compact, $T$ attains its maximum $T_{max}$ and its minimum $T_{min}$.
  Is there a value $T\in]T_{min},T_{max}[$ that is assumed at only one $x_0\in S^2$?   

I think I have to use the fact that the image of a connected space under a continuous function is connected again, but I don't know how. Can you help?

$\textbf{Hint:}$ Assume such a point exists. Remove its pre-image from $\mathbb{S}^2$ and restrict $T$ to this space. Now you use the fact that a continuous image of a connected space is connected.
