(a) find each point of discontinuity. (b) Which of the...

(a) find each point of discontinuity. (b) Which of the discontinuities are removable? not removable? Give reasons for your answers.
$$
f(x)=\left\{\begin{array}{ll}
3-x, & x < 2 \\
2, & x=2 \\
x / 2, & x > 2
\end{array}\right.
$$

Key Concepts

Non-Removable Discontinuity

Non-removable discontinuities are characterized by the failure of the one-sided limits to be equal or the limits not existing at all. Such discontinuities, which include jump discontinuities or essential discontinuities, cannot be corrected simply by redefining the function at that point.

Removable Discontinuity

A removable discontinuity occurs when the limit of the function exists (meaning the left-hand and right-hand limits are equal) at a point, but the function's value at that point is either not defined or does not match that limit. This type of discontinuity can often be 'fixed' by redefining the function at that point to equal the limit.

Continuity and Points of Discontinuity

A function is continuous at a point if the left-hand limit, right-hand limit, and the function's value at that point are all equal. A point of discontinuity occurs when this condition does not hold, meaning there is a break or a jump in the function’s graph.

Piecewise Functions

Piecewise functions are defined by different expressions over various intervals of the domain. They often require careful examination at the boundary points where the expression changes, as these are potential locations for discontinuities.

One-Sided Limits

One-sided limits, including the left-hand limit and the right-hand limit, describe the behavior of a function as the variable approaches a specific point from the left or the right. They are crucial in analyzing continuity at boundary points in piecewise functions to determine if a discontinuity exists.

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