What about continuity and discontinuity of this function?

[vkash]

consider this function $$f(x)=[x[\frac{1}{x}]]$$ ([x] represent greatest integer less than or equal to x or in short GIF )
internal brackets over 1/x and external brackets are around full body of function.
discuss on these points(means either are these correct incorrect)
Statement 1: this function is discontinuous at infinitely points.
Statement 2: this function is discontinuous for infinitely many points for x belongs to (0,1)
I think both statements are correct.
Put x=1/10. It will give 1. but if I put 1/(10.1).1 it will give zero. So it's discontinuous function. similarly we can say that it is discontinuous at infinitely many points between 0 to 1.
If second statement is correct than first will definitely correct...
when result from internal GIF will smaller than 1/x it will give zero, else output should 1.

am i correct?
if not then where am i wrong.

Is there any way to solve such questions on wolframalpha

 

[Office_Shredder]

Put x=1/10. It will give 1. but if I put 1/(10.1).1 it will give zero. So it's discontinuous function and at infinitely many points between 0 to 1.

Ok, let's consider a new function
f(x) = -1010x + 100

f(1/10) = 1
f(1/(10.1)) = 0

Therefore f(x) is a discontinuous function?

 

[vkash]

Ok, let's consider a new function
f(x) = -1010x + 100

f(1/10) = 1
f(1/(10.1)) = 0

Therefore f(x) is a discontinuous function?

I have little cleaner logic here

when result from internal GIF will smaller than 1/x it will give zero, else output should 1.

 

[vkash]

i doesn't understand the graphs represented by wolframalpha?
What is it's values between 0 and 1.
It seems that it is zero for all the values of x (for x in between 0 and 1). But as we can see it is not(ex x=0.1)...