(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.
$$
f(x)=\left\{\begin{array}{ll}
\frac{1}{x-1}, & x < 1 \\
x^{3}-2 x+5, & x \geq 1
\end{array}\right.
$$

Key Concepts

Piecewise Functions

Piecewise functions are functions defined by different expressions over separate intervals of the domain. This concept is crucial because potential discontinuities often occur at the boundaries where the function's definition changes, and analyzing continuity requires considering the different sub-functions on either side of the boundary.

Continuity and Limits

A function is continuous at a point if the right-hand limit, left-hand limit, and the function’s value at that point are equal. Understanding limits is essential to identifying continuity at specific points, especially for piecewise functions, where the behavior as the variable approaches the boundary must be examined from both sides.

Points of Discontinuity

A point of discontinuity is a point in the domain where a function is not continuous. This typically involves situations where the limits from either side of the point do not agree with each other or with the function’s value, indicating a break or gap in the behavior of the function.

Removable Discontinuity

A removable discontinuity occurs when the limit of the function exists at that point (i.e., the left-hand and right-hand limits are equal), but the function is either not defined at that point or defined to have a different value than the limit. This type of discontinuity can be 'fixed' by appropriately redefining the function at that point.

Non-removable Discontinuity

A non-removable discontinuity is one where the left-hand and right-hand limits do not agree or one or both of the limits do not exist, making it impossible to redefine the function at the point to achieve continuity. Examples include jump discontinuities and infinite discontinuities.

Transcript

Please give Ace some feedback