Answer
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The basic types of discontinuity in mathematics are:
1. Jump Discontinuity: This occurs when the limit from the left and the limit from the right exist, but they are not equal. In other words, there's a sudden "jump" in the function's value at the point of discontinuity. An example of this is the function \( f(x) = \begin{cases} 1, & \text{if } x < 0 \\ 0, & \text{if } x \geq 0 \end{cases} \), which has a jump discontinuity at \( x = 0 \).
2. Infinite Discontinuity: This occurs when either or both of the one-sided limits at a point approach infinity. An example is \( f(x) = \frac{1}{x} \), which has an infinite discontinuity at \( x = 0 \).
3. Removable Discontinuity: This occurs when there is a hole in the graph at a particular point, but the limit of the function as it approaches that point exists. This type of discontinuity can be "removed" by redefining the function at that point. An example is \( f(x) = \frac{x^2 - 4}{x - 2} \), which has a removable discontinuity at \( x = 2 \).
In a removable discontinuity, the function is undefined at the point of discontinuity, but the limit of the function as it approaches that point exists and is finite. In the example given, \( f(x) = \frac{x^2 - 4}{x - 2} \), as \( x \) approaches 2, the function approaches 4. However, at \( x = 2 \), the function is undefined due to division by zero. This creates a hole in the graph of the function, which can be "filled in" by defining \( f(2) = 4 \), thus removing the discontinuity.
