Disparity in the definitions of continuity at a point and interval

Question

In trying to deeply understand continuity, I have found (to my limited knowledge) a disparity in the definitions of continuity at points and intervals.

By definition, a point is continuous if it’s left hand limit at x=a, right hand limit at x=a and f(a) are equal.

By definition, an endpoint is continuous if f(a) is equal to the limit which exists for that particular endpoint.

By definition, a function is continuous in an interval if the function is continuous at each point in the interval which in turn, is determined by the above definitions of continuity at points/endpoints.

Now, consider this diagram of a jump discontinuity:

We can state using the above definitions that f(x) is continuous in (- ∞, a] ∪ (a, ∞).

Combining the above ranges we can state that that f(x) is continuous in (- ∞, ∞). But now, again using the same definitions f(x) is discontinuous in (- ∞, ∞).

What have I missed?

Answer 1

The problem is that, for a function defined in $(-\infty, a]$, the limit at $x=a$ does not exist. We might thus make an extension of the definition of continuity to allow for continuity at the boundary. To do that, we demand instead that the left-handed limit is equal to $f(a)$, But note that this in an exception that is only made for points at the boundary. For every other point, we demand that both left and right-handed limits equal $f(a)$. Thus, when you consider the union $(-\infty, a]\cup(a,\infty)$ you cannot treat $a$ as an endpoint anymore. Since "continuity" at endpoints only demands side limits, you will end up with a function that has both a limit to the left of $a$ and a limit to the right of $a$, but they need not be equal.

Answer 2

Another way to see where you are wrong is: You are considering continuity of 3 different functions. The function $f : \mathbb{R} \to \mathbb{R}$ is continuous everywhere except at $a$. The function $f|(-\infty, a]$ is continuous everywhere (via extending the definition of continuity to endpoints) and so is the functions $f|(a, \infty)$. The function $f$ is defined on $\mathbb{R}$ and therefore, even if you write $\mathbb{R} = (-\infty, a] \cup (a, \infty)$, $a$ is not an endpoint of the domain. Hence, we can not use the extended continuity definition in this way and say that $f$ is continuous on $(-\infty, a]$, because it is not continuous at $a$, which is not an endpoint, the right-sided limit can be calculated and the limit at $a$ doesn't exist.

Small remark for clarity: Continuity in an interval means continuity in the domain at every point of the interval. Not continuity in that interval at every point of the interval.