RR SolutionThe function h will not be continuous on all the

Solution

The function h will not be continuous on all the numbers which are of the form, 1/3 * (natural number)

Let the natural number be 1

function at x=1+h, [[3x]] = [[3 * 1/3 * (1+h)]] = [[1+h]] = [1] = 1

function at x=1, [[3x]] = [[3 * 1/3 * (1)]] = [[1]] = [1] = 1

function at x=1+h, [[3x]] = [[3 * 1/3 * (1-h)]] = [[1-h]] = [0] = 0

Hence the function is discontinuous, since left hand limit is not equal to function\'s limit and right hand limit

Therefore function is continous at all R - {1/3*[N]}