## Calculus, 10th Edition (Anton)

$$f(x)=\frac{3}{x}+\frac{x-1}{x^{2}-1}$$ values of $x$, at which $f$ is not continuous are $x = 0$, $x = 1$ and $x = -1$
$$f(x)=\frac{3}{x}+\frac{x-1}{x^{2}-1}$$ suppose $$f(x)=g(x)+h(x)$$ such that $$g(x)= \frac{3}{x}, \quad \quad h(x)=\frac{x-1}{x^{2}-1}$$ The function $g(x)=\frac{3}{x}$ being graphed is a rational function, and hence is continuous at every number except $x=0$. Now consider the other function $$x^{2}-1=0$$ yields discontinuities at $x = 1$ and at $x = -1$. Therefore the function $$h(x)=\frac{x-1}{x^{2}-1}$$ is continuous for all real numbers $x$ except $x = 1$ , $x = -1$ Thus $$f(x)=\frac{3}{x}+\frac{x-1}{x^{2}-1}$$ is continuous for all real numbers $x$ except $x = 0$, $x = 1$ and $x = -1$.