Is a function defined on integers continuous or not??

Answer 1

The answer is Yes, the function is continuous. At least from a mathematical point of view and if you mean a function which maps integers to say the real numbers.

One way to see this is to use the precise definition of continuity.

I wish to give at least a rough picture of that. Take any integer and call it z. To check if the function is continuous at this integer z, you need to play the following game: I give you a real number, say 0.5.

You now need to find another real number, call it h. This h needs to have the property that if you pick another integer n whose distance to z (the integer we fixed above) is less than h, than the distance of their images under our function (i. e. the absolute value of f(z) - f(n)) has to be less than my real number, that is less than 0.5.

Now, since our function is defined on the integers, you can just pick 1 as your number to respond to my number. Why? Because the only number within a distance of less than 1 from z is just z and f(z) - f(z) = 0 which is less than 0.5.

Answer 2

- Will the function accept any real number within its input domain?
- Is the slope of the output continuous within the output range (no gaps, jumps, sharp angles, or invalid results)?

Yes to both questions indicates the function is continuous.
No to either question indicates it is discrete.