Answer
$y=|x-1|+\sin x$ is continuous on $(-\infty,\infty)$
Work Step by Step
$$y=|x-1|+\sin x$$
- Domain: $(-\infty,\infty)$
1) Examine $f(x)=|x-1|$:
$\lim_{x\to c}|x-1|=|c-1|$ for all $x\in(-\infty,\infty)$
So the function $f(x)=|x-1|$ is continuous on $(-\infty,\infty)$.
2) Examine $g(x)=\sin x$
$\lim_{x\to c}\sin x=\sin c$ for all $x\in(-\infty,\infty)$
So the function $g(x)=\sin x$ is continuous on $(-\infty,\infty)$.
According to Theorem 8, the sum of two continous function at $x=c$ is also continous at $x=c$.
Therefore, $y=|x-1|+\sin x$ is continuous on $(-\infty,\infty)$
