The greatest integer, function, represented by $$f(x)=[[x]]$$ is continuous:

**a. everywhere, all real numbers**

**b. on its domain**

**c. only where x is an integer**

**d. none of these**

**------------------**

Determine the value of** $$\lim_{x\rightarrow1} [ [x] ]$$**

**a. 0**

**b. DNE**

**C. 1**

**D. None of these**

**-------------------------------------------**

I'm going to try and look inside my textbook as well.

chilledz3non
Sep 20, 2014

#7**+10 **

Ok. It's the largest integer less than or equal to x, whereas I assumed it was the smallest integer greater than or equal to x. The graph will look pretty much the same though, except it will be shifted along the x-axis by one unit!

And your limit calculations are correct chilledz3non.

Alan
Sep 21, 2014

#2**+8 **

I'm sorry I have not seen questions like these before.

I do not know the notation [[x]] It is a mystery to me?

Melody
Sep 20, 2014

#3**+8 **

I've taken this notation to mean the nearest integer that is greater than or equal to x.

e.g. if x = 2.1 then [[x]] = 3

if x = 1 then [[x]] = 1

if x = -1.5 then [[x]] = -1

Alan
Sep 20, 2014

#5**+3 **

So judging from your graph Allen, I'm going to say that $$f(x)=[[x]]$$ is **not **continuous anywhere, since you have to lift your pencil to draw the graph. Am I right or wrong?

chilledz3non
Sep 20, 2014

#6**+8 **

I finally found it in my textbook, but there is only one example.

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I also did some work to prove that the limit does not exist, but I'm not sure if it is correct.

chilledz3non
Sep 20, 2014

#7**+10 **

Best Answer

Ok. It's the largest integer less than or equal to x, whereas I assumed it was the smallest integer greater than or equal to x. The graph will look pretty much the same though, except it will be shifted along the x-axis by one unit!

And your limit calculations are correct chilledz3non.

Alan
Sep 21, 2014

#8**+5 **

So it is a "Floor" function I think.

I think the answer is "none of these"

The function is continuous for every x that is NOT an integer. That's what I think anyway.

Melody
Sep 21, 2014

#9**0 **

Since I solved the limit and it gave me two different answers for the left side and right side I chose "DNE". Could that be correct as well?

chilledz3non
Sep 21, 2014