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If there is a closed conic function $f(x)$ whose range is real numbers, then is $f(x) \ge 0$ for all $x$? If not, can some simple example which can be easily visualized be given?

Definition: a function is closed conic if its epigraph is closed conic set i.e. it contains all of its limit points and conic combination of any two elements belongs to that set. Conic combination of $u$ and $v$ is $xu+yv$, where $x,y≥0$?

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    Could you please say what a conic combination of $u$ and $v$ is? Is that $xu + yv$, where $x, y \geq 0$? – 2017-02-18
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    Clarified that point. – 2017-02-19

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