Question

If f(x)=[x sinπx], (where [.] denotes the greatest integer function) then f(x) is

• A. Continuous in (–1, 1)
• B. Differentiable at x = –1
• C. Differentiable at x = 1
• D. None of these

Answer 1

Answer: (A)

Continuous in (–1, 1)

Answer 2

By the definition of [x], it is obvious that f(x) = [x sin πx] = 0 when –1 ≤ x ≤ 1 and

f(x) = [x sinπx] = – 1 when 1 < x < 1 + h, (h small) Thus f(x) is constant and equal to 0 in [–1, 1] and hence f(x) is continuous and differentiable in (– 1, 1).

At x = 1, clearly f(x) is discontinuous since $\lim _ { x \rightarrow 1 ^ { + } }$f(x) = – 1 and $\lim _ { x \rightarrow 1 ^ { - } }$f(x) = 0