Let $\mathbb{R}$ denote the set of all real numb... | Question & Solution

Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to\mathbb{R}$ be an arbitrary function and let $g:\mathbb{R}\to\mathbb{R}$ be the function defined by $$g(x)=x f(x),\quad \forall x\in\mathbb{R}.$$ Then which of the following statements is (are) TRUE?

The function $g$ is always continuous at $x=0$

If $f$ is continuous at $x=0$, then $g$ is differentiable at $x=0$

If $g$ is differentiable at $x=0$, then $f$ is continuous at $x=0$

If $g$ is differentiable at $x=0$, then $\lim_{x\to0} f(x)$ exists

Statement (A) is false because g(0)=0·f(0)=0 only if g is defined that way, and continuity is not guaranteed for arbitrary f. For (B), if f is continuous at 0, then f(0) exists and is finite. Hence $g'(0)=\lim_{h\to0}\frac{hf(h)-0}{h}=\lim_{h\to0}f(h)=f(0)$, so g is differentiable at 0. Thus (B) is true. For (C), consider a function f that is not continuous at 0 but has a finite limit as x→0; then g may still be differentiable. Hence (C) is false. For (D), if g is differentiable at 0, then $\lim_{h\to0}\frac{g(h)-g(0)}{h}=\lim_{h\to0}f(h)$ exists, implying that $\lim_{x\to0}f(x)$ exists. Therefore (D) is true.

Comments and discussions will be available soon.

Question

Detailed Solution

Discussion

Appears in Collections

Related Questions