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## Homework Statement

If [itex]f:\mathbb{R}\to\mathbb{R}[/itex] and [itex]g:\mathbb{R}\to\mathbb{R}[/itex] are continuous functions, give examples to show that the set [itex]\{ (f(x),g(x)) : x\in\mathbb{R} \}[/itex] might or might not be closed in [itex]\mathbb{R}^2[/itex].

## The Attempt at a Solution

Letting [itex]f(x)=g(x)=0[/itex] gives the set equal to [itex]\{ (0,0) \}[/itex], a singleton and singleton sets are closed.

What functions would make the set open?