Define f(x) as the product of two real functions f_1(x)=x, xR, and f_2(x)=cases1x, & x00, …

Question

Define $f(x)$ as the product of two real functions $f_1(x)=x,\ x\in\mathbb{R}$, and $f_2(x)=\begin{cases}\sin\frac1x, & x
e0\0, & x=0\end{cases}$ as follows: $f(x)=\begin{cases}f_1(x)f_2(x), & x
e0\0, & x=0\end{cases}$.

Statement 1: $f(x)$ is continuous on $\mathbb{R}$.

Statement 2: $f_1(x)$ and $f_2(x)$ are continuous on $\mathbb{R}$.

Accepted Answer

$f(x)=x\sin\frac1x$ (for $x
e0$) is continuous at $0$ (bounded times vanishing factor) --- Statement 1 true. But $f_2(x)=\sin\frac1x$ oscillates and is discontinuous at $x=0$ --- Statement 2 false.

Solutions · 1