Proving Continuity at 2 points ONLY

Question

I have a piecewise function:

f(x) = 0 if x is rational

= x(x-1) if x is irrational

I want to prove that this function is continuous at exactly two points

Answer 1

x=0 and x=1

f(0) = 0 since 0 is a rational number 0 = 0/1 a ratio of two integers

f(1) = 0 since 0 is a rational number

as x approaches 0 as an irrational number, f(x) approaches 0(0-1) = 0

the limit of f(x) as an irrational number = f(x) where x is a rational number

whether x approaches 0 as a negative number or a positive number

as x approaches 1 as an irrational number f(x) approaches 1(1-1) = 0

the limit of f(x) with x as an irrational number = f(x) where x is a rational number

as x approaches 1 from the left or right.