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Consider $f$ is continuous. We know that if $f$ is monotonic then $f$ is also one to one But can we conclude if $f$ is one to one then $f$ is also monotonic ? (If it's true how we can prove it?)

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    No, the injective property is not enough. Hint: think of a discontinuous example. – 2017-02-03
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    $f$ is a function from what into what? Consider a non continuous example... – 2017-02-03
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    $f$ is continuous – 2017-02-03
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    this depends on the domain (consider $f$ defined on $(0,1)\cup (2,3)$) – 2017-02-03
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    Is $f$ a function from a subset of the reals to the reals and is the domain connected? – 2017-02-03

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If $f$ is continuous on a connected interval and one-to-one, it will be monotonic. If not there are $af(c)$ or similar numbers with reverse inequalities. Now by the mean value theorem each $\max\{f(a), f(c)\}

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    Can you explain the second part of your answer with an example ? – 2017-02-03
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    @S.H.W if $f$ is continuous, $a$ d\in (a,b)$ such that $f(d) = x$. The same applies on the interval $(b,c)$ – 2017-02-03