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I need to find the continuity of the functions: $f(x)=[x]$, where [x] is the integer part and $f(x')=${x'}, where {x'} is the fractional part. I though about using series but if you could show me how or any other method it would be great.

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    Have you tried drawing these functions to see how they behave? – 2017-01-22
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    Yeah I know that they are not continuous on Z using that method. But I want to know one other than that. – 2017-01-22

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Here is a drawing of the floor function:

enter image description here

and here is one of the fractional part function (depending of your definition of it):

enter image description here

Now that you have an intuition about these functions, can you find where they are discontinuous?

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    Well I knew he was discontinuous on Z already by drawing. But I want to see other method. Like just writing or using series. – 2017-01-22
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    @OvyOvy Try limits :) – 2017-01-22
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    I would if I knew how. What limits? – 2017-01-23
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    $\lim_{x\to a^-} f(x)$ and $\lim_{x\to a^+} f(x)$, and try to see if those limits equal $f(a)$ – 2017-01-23
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    Well it depends. If a is integer then they are different. If not, then they are equal. – 2017-01-23
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    @OvyOvy There you go, and here is how you find the continuity. – 2017-01-23
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    Yeah but I found it by guessing. Isn't there any method that works for any function? – 2017-01-23
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    @OvyOvy There is not. – 2017-01-23
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    Ok then. Thanks. – 2017-01-23