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Question:

Let f, g: R → R be two functions defined by f(x) = xsin(1/x) for x ≠ 0, f(x) = 0 for x = 0, and g(x) = xf(x). Statement I: f is a continuous function at x = 0. Statement II: g is a differentiable function at x = 0. Choose the correct statement from the following options: Both statements I and II are false. Both statements I and II are true. Statement I is true, statement II is false. Statement I is false, statement II is true.

Both statements I and II are false.

Both statements I and II are true.

Statement I is true, statement II is false.

Statement I is false, statement II is true.

Solution:

RHL = limh→0+ hsin(1/h) = 0 × finite number = 0
LHL = limh→0− (−h)sin(1/h) = limh→0− −hsin(1/h) = 0 × finite number = 0
f(0) = 0
Hence, f is continuous at x = 0
g(x) = xf(x) = x²sin(1/x).
Clearly, g(0) = 0
g'(x) = limh→0 g(x+h) − g(x)/h
g'(0) = limh→0 (h)²sin(1/h)/h = 0 (finite)
Hence g(x) is differentiable at x = 0