## University Calculus: Early Transcendentals (3rd Edition)

A function being continuous on an interval means that it is continuous at every point on that interval. And a function that is not continuous on its entire domain can still be continuous on selected intervals within that domain. For example, take this function: $f(x)=2$ for $x\ne0$ and $f(x)=0$ for $x=0$ - Domain: $R$ - We can easily that $\lim_{x\to0}f(x)=\lim_{x\to0}2=2$ Yet $f(0)=0$ So since $\lim_{x\to0}f(x)\ne f(0)$, the function is not continuous at $x=0$, so it is not continuous on its entire domain. - However, for all the points $x=c$ on the intervals $(-\infty,0)$ and $(0,\infty)$, $\lim_{x\to c}f(x)=f(c)=2$. This means the function $f(x)$ is continuous at every point on $(-\infty,0)$ and $(0,\infty)$, meaning that even though $f(x)$ is not continuous on the entire domain, it is still continuous on these intervals $(-\infty,0)$ and $(0,\infty)$ within the domain.