Game Theory: Are the payoff functions πi continuous?

[vonanka]

1. How do I show that the payoff function πi isn´t continuous? Why do best replies not always exist?
   
   
   

 

[mfb]

How do I show that the payoff function πi isn´t continuous?

In the same way you show that a function is not continuous for (nearly) every other function. Find a point where it is not continuous and prove that the function does not satisfy the condition for continuity there.

Is this a homework problem?

 

[vonanka]

In the same way you show that a function is not continuous for (nearly) every other function. Find a point where it is not continuous and prove that the function does not satisfy the condition for continuity there.

Is this a homework problem?

Okey, but what if the interval was extremely large. Is there a way to find the non continuous part in a smart algorithmic way?
It´s a old exam without answers. Really need a explanation here. Thanks.

 

[mfb]

If you have a piecewise definition, the borders between the pieces are always obvious places to check.
The pieces itself can be discontinuous as well, but if they are simple polynomials like here you know they are continuous.