Find a function with certain property

Question

I am searching for an example of the function with the following property.

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For given function $f : \mathbb{R}\rightarrow\mathbb{R}$, both $e^{x}f(x)$ and $e^{-f(x)}$ are monotonically decreasing.

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I think since $e^{-f(x)}$ is monotonically decreasing, so that $f(x)$ should increase but it does not easily come up with that kind of function which makes $e^{x}f(x)$ monotonically decreasing. How can I find the example?

EDIT: The original question is after we assume the function with that kind of property, prove the function $f$ is continuous.

Answer 1

How about $f(x)\equiv -1$? Any constant function is monotonically increasing and monotonically decreasing!