1. Tardigrade
  2. Question
  3. Mathematics
  4. Let f, g: R → R be two functions defined by f(x) = begincases x sin ( (1/x) ), x ≠ 0 0 , x = 0 endcases , and g(x) = x f(x) Statement I: f is a continuous function at x = 0. Statement II: g is a differentiable function at x = 0.

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Q. Let f, g: R $\to$ R be two functions defined by $f(x) = \begin{cases} x \sin \left( \frac{1}{x} \right), & x \neq 0 \\ 0 , & x = 0 \end{cases}$ , and g(x) = x f(x)
Statement I: f is a continuous function at x = 0.
Statement II: g is a differentiable function at x = 0.

Continuity and Differentiability

Solution:

Correct answer is (b) Both statement I and II are true