We may show that a function is continuous over an interval $[a,b]$ by applying
$$\lim_{x \to a+} f(x) = f(a) \quad \text{and} \quad \lim_{x \to b−}f(x) = f(b)$$
But what about on an interval $(-\infty, 3]$ ? How can the definition of 'continuous on an interval' above be applied with an infinite value, seeing as infinity cannot be plugged in to the expression?
Best Answer
Actually, to show that a function is continuous on an interval you need to show that the limits agree at every point in the interval: $$ \lim_{x\to c}f(x)=f(c),\quad c\in (a,b), $$ in addition to checking the limits at the endpoints as you have written.
For a semi-infinite interval like $(-\infty,3]$, you still need to check the limit at each point in the interior (i.e. all $c\leq 3$ in this case), but there is no limit to check at $-\infty$. You would still check the upper limit at $3$, of course.