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Be $x=(x_{n})$,$y=(y_{n})\in \mathbb{R}^\omega$, be $f\colon [0,1]\subseteq\mathbb{R}\rightarrow \mathbb{R}^\omega$ and $f(t)=(1-t)x_{n}+ty_{n}$.

For $\mathbb{R}^\omega$ with the box topology, show that $f$ is continuous if only if $\exists N\in \mathbb{N}$ that $\forall_{n \geq N} x_{n}=y_{n}$

$\Leftarrow$ no problem,

$\Rightarrow$ I have problems, i think in use continuous properties, like $f(\overline{A})\subseteq\overline{f(A)}$ but dont result.

Any help is appreciated.

$$\mathbb{R}^\omega=\prod_{n=1}^{\infty}{\mathbb{R}}$$

$\mathbb{R}$ usual topology

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