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Show that if $N=2m+1$, then $U_\epsilon(C)$ is dense in $\mathcal{C}(X,\mathbb{R}^N)$.

I am given the following hint:

Given $f\in \mathcal{C}(X,\mathbb{R}^N)$ and $\delta,\epsilon >0$ choose $g:C\to \mathbb{R}^N$ so that: $d(f(x),g(x))<\delta$ for $x\in C$, and $\Delta(g)<\epsilon$. Extend $f-g$ to $h: X \to [-\delta,\delta]^N$ using the Tietze theorem.

Where does he use the fact that $N=2m+1$?, we have: $f|_C = g+ h|_C$, so $\Delta(f|_C) = \Delta(g+h|_C)< \epsilon + (2\delta)^N$

We need to show that $f\in U_\epsilon(C)$ or that it's a limit point of $U_\epsilon(C)$, how exactly I don't see it.

Thanks in advance.