91Ó°ÊÓ

Piecewise-defined functions are a type of function that have different expressions or formulas for different parts of their domain. Each of these pieces applies to a certain interval or condition. For example, in our problem, the function \( g(x) \) is defined differently depending on whether \( x \) is greater than or equal to zero or less than zero. This can be particularly useful in modeling situations where different rules apply to different scenarios.

• For \( x \geq 0 \), \( g(x) = x^{2/3} \)
• For \( x < 0 \), \( g(x) = x^{1/3} \)

Understanding the conditions under which each piece of the function applies is crucial when analyzing the overall behavior of the function, including its continuity and differentiability at specific points.

Continuity at a point means that small changes in the input around that point result in small changes in the output. Mathematically, a function \( f(x) \) is continuous at a point \( x = c \) if the following condition is satisfied:
\[ \lim_{{x \to c^-}} f(x) = f(c) = \lim_{{x \to c^+}} f(x). \]
In simple terms, the left and right limits at that point are equal to the function’s value at the point. For piecewise-defined functions, checking continuity involves ensuring that the pieces connect neatly without any jumps or gaps at the boundaries.
For this particular problem, checking continuity at \( x = 0 \) would require ensuring that the left and right limits of \( g(x) \) as \( x \) approaches 0 from both sides equal \( g(0) \). However, differentiability requires more than just continuity; it requires the function to be smoothly changing at that point.

The left-hand derivative of a function at a point \( x = c \) refers to the derivative that is approached as \( x \) approaches \( c \) from the left. It measures the rate at which the function changes as you come from the left side of that point. Mathematically, it is expressed as:
\[ \lim_{{h \to 0^-}} \frac{f(c+h) - f(c)}{h} \]
For our problem, the left-hand derivative of \( g(x) = x^{1/3} \) as \( x \to 0^- \) evaluates to infinity, indicating that the derivative does not exist at \( x = 0 \) from the left. This absence of a finite derivative is one of the signals that a function may not be differentiable at that point.

The right-hand derivative of a function at a point \( x = c \) is the derivative calculated as \( x \) approaches \( c \) from the right. It represents the function's rate of change approaching the point from the positive side. The formula for the right-hand derivative is:
\[ \lim_{{h \to 0^+}} \frac{f(c+h) - f(c)}{h} \]
In this case, the right-hand derivative for \( g(x) = x^{2/3} \) as \( x \to 0^+ \) is also infinite. Like the left-hand derivative, the result shows the derivative does not exist as \( x \to 0 \) from the right. The lack of a finite right-hand derivative paired with the nonexistence of the left-hand derivative concludes that the function is not differentiable at the origin. This illustrates how differentiability requires both directional derivatives to exist and be equal. If one or both are infinite, like in this case, the function is not differentiable at that point.