Answer
All the basic types of discontinuity and removable discontinuity are detailed below.
Work Step by Step
1) The basic types of discontinuity:
- Jump discontinuity: a function "jumping" from one value to another at a point $x=c$ is discontinuous at that point.
Function $f(x)=1.9$ for $x\lt2$ and $f(x)=2.1$ for $x\ge2$
This function "jumps" at $x=2$ from $1.9$ to $2.1$. In mathematical terms, we see that $\lim_{x\to2}f(x)$ does not exist, so the function is discontinuous at $x=2$.
- Infinite discontinuity: the function being undefined at $x=c$ because as $x\to c$, the function approaches $\pm\infty$ has an infinite discontinuity at $c$.
Function $f(x)=1/x$ has an infinite discontinuity at $x=0$ because $\lim_{x\to0}f(x)=\infty$ and does not exist, so it does not pass the continuity test and is not continuous at $x=0$.
- Oscillating discontinuity: the function oscillating too much at a point to have a limit as $x\to c$ has an oscillating discontinuity at $x=c$
Function $f(x)=\sin(1/x)$ (whose graph is shown below) oscillates too much to have a limit as $x\to0$. It, therefore, has an oscillating discontinuity at $x=0$.
2) Removable discontinuity:
A function $f(x)$ has a removable discontinuity at $x=c$ if:
- $f(c)$ and $\lim_{x\to c}f(x)$ both exist.
- $\lim_{x\to c}f(x)\ne f(c)$
This means in theory, we can "remove" the discontinuity simply by bringing the value of $f(c)$ to be equal with the value of $\lim_{x\to c}f(x)$.
For example, take function $f(x)=3/2$ for $x=1$ and $f(x)=1/(x^3+1)$ for $x\ne 1$
Here, $\lim_{x\to 1}f(x)=1/(1^3+1)=1/2$ while $f(1)=3/2$
So as $\lim_{x\to 1}f(x)\ne f(1)$, $f(x)$ has a removable discontinuity at $x=1$ because we can theoretically "remove" the discontinuity by including a value of $f(1)=1/2$ so that $\lim_{x\to 1}f(x)=f(1)$.