For the following exercises, decide if the function...

For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? $$ f(x)=\left\{\begin{array}{ll}{x \sin (x)} & {\text { if } x \leq \pi} \\ {x \tan (x) \text { if } x>\pi}\end{array}, \text { at } x=\pi\right. $$

For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it?
$$
f(x)=\left\{\begin{array}{ll}{x \sin (x)} & {\text { if } x \leq \pi} \\ {x \tan (x) \text { if } x>\pi}\end{array}, \text { at } x=\pi\right.
$$

Key Concepts

Piecewise-Defined Functions

These are functions that have different expressions or rules for different intervals of their domain. Understanding how each piece behaves and how they connect is crucial when analyzing the overall properties of the function, especially in terms of continuity or differentiability.

Continuity at a Point

A function is continuous at a point if the limit from the left and the limit from the right exist, are equal, and equal the function’s value at that point. This concept is central in calculus and is used to ensure there are no 'gaps' or 'jumps' at the point in question.

One-Sided Limits

One-sided limits involve approaching a particular point from only one direction—either from the left or the right. They are essential in the study of piecewise-defined functions, as they help determine the behavior of the function as it approaches the point from different directions.

Types of Discontinuities

Discontinuities are classified based on how the limits approach a point relative to the function’s value at that point. Common types include removable discontinuities (where the limit exists but does not equal the function’s value), jump discontinuities (where the left-hand and right-hand limits are not equal), and essential or infinite discontinuities (where at least one of the one-sided limits does not exist or is unbounded).

Trigonometric Functions in Continuity Analysis

Trigonometric functions, such as sine and tangent, have specific periodic and asymptotic properties that affect their behavior around particular points. Their characteristics must be carefully analyzed, especially near points where the functions might have undefined values or discontinuous behavior, impacting the continuity of a piecewise-defined function.

Please give Ace some feedback