Define fRR by f x x3 if x is rational and f x2 if x is irrat

Define f:R->R by f (x) =x-3 if x is rational and f (x)=2 if x is irrational. Prove that f is continuous at c existing in R if and only if c=5

Solution

f(x) = x - 3, if x is rational

f(x) = 2, if x is irrational

If x is an irrational number, then the value of f(x) will be equal to 2

If we take any rational number close to that irrational number, then the value of f(x) will be (rational number - 3)

In order for the function to be continuous, both the irrational and rational limits must be same

x - 3 = 2

x = 5

Hence the only number for which function is continuous that is c=5