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.

Question

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. (A) Both statement I and II are false (B) Both statement I and II are true (C) Statement I is true, statement II is false (D) Statement I is false, statement II is true

Tardigrade Admin · Accepted Answer

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

Q. Let f, g: R → R be two functions defined by f(x)={xsin(x1​),0,​x=0x=0​ , 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.

Both statement I and II are false

Both statement I and II are true

Statement I is true, statement II is false

Statement I is false, statement II is true

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

Q. Let f, g: R $\to$ R be two functions defined by $f (x) = {x sin (\frac{1}{x}), 0, x \neq = 0 x = 0$ , 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.