Answer
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.
Work Step by Step
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.
