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Is the function from $\mathbb{R}$ to $\mathbb{R}$ that maps every compact set to compact set and connected set to connected set continious?

I know that if the function maps only compact set (respectively connected) to compact (connected) it's not continious. I saw the counterexample. Is it also true when we take both properties?

Thanks in advance.

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    Are you talking about functions from $\mathbb R$ to $\mathbb R,$ or functions from a general topological space to a general topological space, or what? – 2017-01-04
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    Ok. Jimmy R. gave counterexample for general topological spaces. So how is it with function $\mathbb{R}$ to $\mathbb{R}$? – 2017-01-04
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    See [this](http://math.stackexchange.com/questions/220410/a-characterization-of-functions-from-mathbb-rn-to-mathbb-rm-which-are-co). – 2017-01-04
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    Ok. I suck in finding stuff. Thanks. That is the awnser for my question. – 2017-01-04

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