Notice in the above definition of continuity on an interval \((a,b)\) we have carefully avoided saying anything about whether or not the function is continuous at the endpoints of the interval — i.e. is \(f(x)\) continuous at \(x=a\) or \(x=b\text{.}\) This is because talking of continuity at the endpoints of an interval can be a little delicate.
In many situations we will be given a function \(f(x)\) defined on a closed interval \([a,b]\text{.}\) For example, we might have:
For any \(0 \leq x \leq 1\) we know the value of \(f(x)\text{.}\) However for \(x \lt 0\) or \(x \gt 1\) we know nothing about the function — indeed it has not been defined.
So now, consider what it means for \(f(x)\) to be continuous at \(x=0\text{.}\) We need to have
however this implies that the one-sided limits
Now the first of these one-sided limits involves examining the behaviour of \(f(x)\) for \(x \gt 0\text{.}\) Since this involves looking at points for which \(f(x)\) is defined, this is something we can do. On the other hand the second one-sided limit requires us to understand the behaviour of \(f(x)\) for \(x \lt 0\text{.}\) This we cannot do because the function hasn't been defined for \(x \lt 0\text{.}\)
One way around this problem is to generalise the idea of continuity to one-sided continuity, just as we generalised limits to get one-sided limits.
A function \(f(x)\) is continuous from the right at \(a\) if
Similarly a function \(f(x)\) is continuous from the left at \(a\) if
Using the definition of one-sided continuity we can now define what it means for a function to be continuous on a closed interval.
A function \(f(x)\) is continuous on the closed interval \([a,b]\) when
Note that the last two condition are equivalent to