Removable and Jump Discontinuities

Not all discontinuities are the same "kind" — some can be patched by simply reassigning a single function value, while others cannot be patched at all no matter what value is chosen.

Removable discontinuity. Consider $f(x)=\dfrac{\sin x}{x}$, undefined at $x=0$ (failure type (1) of Section 9.3). Even though $f(0)$ does not exist, $\lim_{x\to0}\dfrac{\sin x}{x}=1$ genuinely exists. If a new function $h$ is defined to agree with $f$ everywhere except that it is explicitly assigned the value $h(0)=1$ — exactly the limiting value — then $h$ becomes continuous at $0$: the "hole" in the graph is simply filled in with the single point the limit was always pointing to. Because the discontinuity can be eliminated this way, by redefining a single point, it is called removable.

Definition 9.10. $f$, defined on an interval $I$, has a removable discontinuity at $x_0\in I$ if there exists a function $h:I\to\mathbb R$ with $h(x)=f(x)$ for all $x\ne x_0$, $h(x_0)=\lim_{x\to x_0} f(x)$, and $h$ continuous at $x_0$. (Necessarily, for this to even be possible, $\lim_{x\to x_0}f(x)$ must exist as an ordinary two-sided limit in the first place — it is exactly the value that gets assigned to patch the hole.)

Jump discontinuity. Revisit the tomato-cost function $C(x)$ from Example 9.38: at $x=100$, both one-sided limits exist ($16$ from the left, $14$ from the right) but they simply disagree, giving a finite jump of height $\lim_{x\to100^-}C(x)-\lim_{x\to100^+}C(x)=16-14=2$. Unlike the removable case, no single value assigned at $x=100$ could fix this — reassigning $C(100)$ only relocates where the dot sits on the graph; it can never make the two approaching branches meet. This is a fundamentally different, unpatchable kind of discontinuity:

Definition 9.11. $f$, defined on interval $I$, has a jump discontinuity at $x_0\in I$ if $f$ is defined at $x_0$ and both one-sided limits $\lim_{x\to x_0^-}f(x)$ and $\lim_{x\to x_0^+}f(x)$ exist, but $\lim_{x\to x_0^-}f(x)\ne\lim_{x\to x_0^+}f(x)$.

Distinguishing the two. The dividing line between "removable" and "jump" is exactly whether the two-sided limit exists at all: a removable discontinuity has a perfectly good (single) two-sided limit that the function value simply fails to match, or fails to have at all; a jump discontinuity has two perfectly good but different one-sided limits, so no single two-sided limit exists to be assigned in the first place. …