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A piecewise function is a type of function defined by different expressions depending on the value of the independent variable, often denoted as "x". In simple terms, this means that instead of having a single algebraic expression, a piecewise function has multiple expressions. Each part of the piecewise function applies to a certain interval of the domain.

For example, the function provided in the exercise is defined by two different expressions:

• \( \sin(|x|) \) when \( x \le 0 \)
• \( 1-x \) when \( x > 0 \)

This means \( f(x) \) uses the expression \( \sin(|x|) \) for all \( x \) values less than or equal to zero, and the expression \( 1-x \) for \( x \) values greater than zero. Such functions allow more flexibility and are quite useful in modeling scenarios where a function's behavior needs to change at certain points.

Understanding piecewise functions is crucial before assessing continuity or differentiability, as each piece can behave differently.

Continuity of a function at a certain point means the function behaves smoothly without any breaks or jumps at that point. Mathematically, for a function \( f(x) \) to be continuous at a point \( c \), the following must hold true:

• The function \( f \) is defined at \( c \), that is, \( f(c) \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( c \) from both directions (left and right) exists.
• The limit value and the function value at \( c \) must be the same: \( \lim_{x \to c} f(x) = f(c) \).

If any of these conditions are not met, the function \( f(x) \) is not continuous at that point.

In the given exercise, \( f(x) \) is not continuous at \( x = 0 \) since the left-hand limit (approaching from negative values of \( x \)) and the right-hand limit (approaching from positive values of \( x \)) are not equal. Thus, \( f(x) \) has a jump at \( x = 0 \). This jump indicates discontinuity, which is crucial in understanding the differentiability of a function.

The limit of the difference quotient is a fundamental concept in calculus related to the definition of a derivative. To assess the differentiability of a function at a point \( c \), one computes the derivative using the difference quotient:\[ \lim_{h \to 0} \frac{f(c+h) - f(c)}{h} \]This expression represents the rate of change of \( f(x) \) as \( x \) approaches \( c \). For a function to be differentiable at a point, the limit must exist and be the same when approached from both sides (left-hand and right-hand limits).

Continuity plays a vital role here. A function that is not continuous at a point cannot have a well-defined derivative there because the limits from both sides do not match. For the exercise provided, because \( f(x) \) is not continuous at \( x = 0 \), no matter how we attempt to calculate the difference quotient, we end up without a definitive answer for the derivative at that point.

A discontinuous function is one that has at least one point where it is not continuous. This could be due to a jump, a hole, or an infinite discontinuity at those points.

Let's break down some common types of discontinuity:

• Jump Discontinuity: Occurs when the left-hand limit and the right-hand limit exist but are not equal. This is what we observe at \( x = 0 \) for the provided function, as the limit from the left is different from the limit from the right.
• Removable Discontinuity: Occurs when the function is not defined at a point, but a limit exists from both directions and is the same. Often, it can be "fixed" by adjusting the function’s definition at that point.
• Infinite Discontinuity: Occurs when the function approaches infinity at a specific point, such as asymptotes in rational functions.

Identifying types of discontinuity helps in analyzing a function's behavior and is critical to understanding and applying concepts like differentiability. In fact, as shown in the exercise, discontinuity directly impacts a function's differentiability at a given point.