Consider the function f : R → R defined by f (x)=1 if x∈ Q, f (x)=0 if x∈ R/Q, where is f continuous? be sure to prove your assertion

The function you've described is known as the Dirichlet function. It's a classic example of a function that is discontinuous at every point in its domain.

Here's a step-by-step proof:

1. Let's first understand the function. The function f(x) is defined to be 1 for all rational numbers (numbers that can be expressed as a quotient of two integers) and 0 for all irrational numbers (numbers that cannot be expressed as a quotient of two integers).

2. We want to show that this function is discontinuous at every point in its domain. To do this, we'll use the definition of continuity. A function f(x) is continuous at a point x=a if the limit as x approaches a of f(x) equals f(a).

3. Let's consider an arbitrary point a in the domain of f. Without loss of generality, we can consider two cases: a is rational and a is irrational.

4. If a is rational, then f(a) = 1. However, for any neighborhood around a, no matter how small, there will always be irrational numbers. For these numbers, f(x) = 0. Therefore, the limit as x approaches a of f(x) does not exist, so f is not continuous at a.

5. If a is irrational, then f(a) = 0. However, for any neighborhood around a, no matter how small, there will always be rational numbers. For these numbers, f(x) = 1. Therefore, the limit as x approaches a of f(x) does not exist, so f is not continuous at a.

6. Since a was an arbitrary point in the domain of f, we can conclude that f is discontinuous at every point in its domain.