A function which is continuous on Z only

In summary, the conversation discusses finding an example of a function defined on R which is continuous on Z only. The participants also discuss finding a function that is continuous at the origin only and repeating it. The main focus is on modifying a function that is discontinuous everywhere to become continuous at the integers.
  • #1
HappyN
16
0
I have spent ages on this final part of a question but don't seem to be going anywhere - any help would be greatly appreciated!

Given a function f:R->R let X be the set of all points at which f is continuous.
Find an example of a function defined on R which is continuous on Z only.
 
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  • #2
Hi HappyN! :smile:
HappyN said:
Find an example of a function defined on R which is continuous on Z only.

Find a function continuous at the origin only, and then repeat it. :wink:
 
  • #3
tiny-tim said:
Find a function continuous at the origin only, and then repeat it. :wink:

Thanks, I'm not quite sure what you mean by 'repeat it' though? If a function is only continuous at the origin surely it is not continuous on all Z?
 
  • #4
So have you found a function continuous at the origin only?
 
  • #5
HappyN said:
I have spent ages on this final part of a question but don't seem to be going anywhere - any help would be greatly appreciated!

Given a function f:R->R let X be the set of all points at which f is continuous.
Find an example of a function defined on R which is continuous on Z only.

Start with a function that is discontinuous everywhere and see if you can modify it so that it becomes continuous at the integers but nowhere else.
 
  • #6
Landau said:
So have you found a function continuous at the origin only?

I got f(x)={ x x € Q
{-x otherwise
not sure if that is right though?
 
  • #7
HappyN said:
I got f(x)={ x x € Q
{-x otherwise
not sure if that is right though?

what happens at integers other than zero?
 
  • #8
HappyN said:
I got f(x)={ x x € Q
{-x otherwise
not sure if that is right though?

ok, now chop out a bit round the origin, and keep copy-and-pasting it :wink:
 

FAQ: A function which is continuous on Z only

What is a function that is continuous on Z only?

A function that is continuous on Z only means that it is only continuous at integer values of x, and is not continuous at any non-integer values. This means that the function has a "jump" or "break" in its graph at non-integer values.

How can you determine if a function is continuous on Z only?

To determine if a function is continuous on Z only, you can visually inspect its graph and look for any "jumps" or "breaks" at non-integer values. You can also use the epsilon-delta definition of continuity to prove that the function is not continuous at non-integer values.

Can a function be continuous on Z only and still be considered a "good" or "useful" function?

Yes, a function can still be considered "good" or "useful" even if it is only continuous on Z. These types of functions are often used in discrete mathematics and computer science, and can still provide valuable information and insights.

What is an example of a function that is continuous on Z only?

One example of a function that is continuous on Z only is the floor function, denoted as f(x) = ⌊x⌋. This function returns the largest integer less than or equal to x, and is only continuous at integer values of x. At non-integer values, there is a "jump" or "break" in the graph.

How is a function that is continuous on Z only different from a function that is continuous on all real numbers?

A function that is continuous on Z only is different from a function that is continuous on all real numbers because it is not continuous at non-integer values. A function that is continuous on all real numbers has a smooth and unbroken graph, while a function that is continuous on Z only has "jumps" or "breaks" at non-integer values.

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