What is the continuity of these functions?

Question

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.

Have you tried drawing these functions to see how they behave? — Ofir, Jan 22 '17 at 21:30
Yeah I know that they are not continuous on Z using that method. But I want to know one other than that. — Ghost, Jan 22 '17 at 21:34

score 1 · Accepted Answer · answered Jan 22 '17 at 21:34

1

Here is a drawing of the floor function:

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

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

answered Jan 22 '17 at 21:34

E. Joseph

14,843

Well I knew he was discontinuous on Z already by drawing. But I want to see other method. Like just writing or using series. – Ghost Jan 22 '17 at 21:35
@OvyOvy Try limits :) – E. Joseph Jan 22 '17 at 22:01
I would if I knew how. What limits? – Ghost Jan 23 '17 at 07:57
$\lim_{x\to a^-} f(x)$ and $\lim_{x\to a^+} f(x)$, and try to see if those limits equal $f(a)$ – E. Joseph Jan 23 '17 at 08:26
Well it depends. If a is integer then they are different. If not, then they are equal. – Ghost Jan 23 '17 at 08:27
@OvyOvy There you go, and here is how you find the continuity. – E. Joseph Jan 23 '17 at 08:29
Yeah but I found it by guessing. Isn't there any method that works for any function? – Ghost Jan 23 '17 at 08:29
@OvyOvy There is not. – E. Joseph Jan 23 '17 at 08:30
Ok then. Thanks. – Ghost Jan 23 '17 at 08:31

What is the continuity of these functions?

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