There are 2 possible ways of choosing an interval of cot x graph for obtaining the graph of arccot x. Why do we choose the interval (0,$\pi$) instead of the first choice? This may be because the graph is continous at that interval,but the graphs of arcsec x and arccosec x aren't continous too and we don't generally face problems working with them, so what is the reason in this case? 
Correct graph of arccot x
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$\begingroup$
trigonometry
inverse-function
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0The official one is the second one. – 2017-02-18
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0Why do you assume there is a reason? – 2017-02-18
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1It is indeed because we prefer continuous functions where possible. Your comparison to arcsec and arccsc are not apropo because neither of those functions allows a continuous expression. No matter how they are defined, it has to skip over that gap from $-1$ to $1$. Continuity cannot be achieved. But arccot can be chosen continuously, so we do. – 2017-02-18