Problem 4 - A Primer of Real Analysis

Question

Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by $$ f(x)=\left\\{\begin{array}{ll} x^{2}, & \text { if } x<0 \\ x^{3}, & \text { if } x \geq 0 \end{array}\right. $$ Is \(f\) differentiable at \(0 ?\)

Step-by-Step Solution

Answer

Yes, \(f\) is differentiable at 0.

Step 1: Definition of Differentiability

A function \(f\) is differentiable at a point \(a\) if the limit \(\lim_{x \to a}\frac{f(x)-f(a)}{x-a}\) exists.

Step 2: Evaluate \(f(0)\)

Calculate \(f(0)\) using the piecewise definition of \(f\). Since \(x = 0\), we use \(f(x) = x^3\). Thus, \(f(0) = 0^3 = 0\).

Step 3: Calculate the Left-Hand Limit

For \(x < 0\), \(f(x) = x^2\). Compute the limit \(\lim_{x \to 0^-}\frac{f(x)-f(0)}{x-0} = \lim_{x \to 0^-}\frac{x^2}{x} = \lim_{x \to 0^-} x = 0\).

Step 4: Calculate the Right-Hand Limit

For \(x \geq 0\), \(f(x) = x^3\). Compute the limit \(\lim_{x \to 0^+}\frac{f(x)-f(0)}{x-0} = \lim_{x \to 0^+}\frac{x^3}{x} = \lim_{x \to 0^+} x^2 = 0\).

Step 5: Compare the Limits

The left-hand limit and right-hand limit at \(x=0\) are both 0. Therefore, \(\lim_{x \to 0}\frac{f(x)-f(0)}{x-0} = 0\) exists.

Step 6: Conclusion on Differentiability

Since both the left-hand and right-hand limits exist and are equal, \(f\) is differentiable at \(x=0\).

Key Concepts

Piecewise functions

A piecewise function is a type of function that is defined by different expressions depending on the input value. Each "piece" or part of the function applies to a certain interval of the domain. This can be helpful in modeling real-world situations where the behavior of a function changes at different intervals.

• For example, the function given in the original exercise is defined as \( f(x) = x^2 \) when \( x < 0 \), and as \( f(x) = x^3 \) when \( x \geq 0 \).
• Such functions are often used to model situations where a different rule needs to be applied depending on certain conditions.

Understanding how to work with piecewise functions involves knowing which expression to use based on the input value. When dealing with them, one of the key steps is carefully identifying the intervals and the correct formula that applies to each interval. This is an important concept when analyzing limits and continuity, as the behavior of the function can change drastically at the transition points.

Left-hand limit

The left-hand limit refers to the value that a function approaches as the input approaches a particular point from the left-hand side. This means that we're considering values just below the point in question.

• Mathematically, for a point \( a \), the left-hand limit is denoted as \( \lim_{x \to a^-} f(x) \).
• In the context of the original exercise for the function \( f \) at \( x = 0 \), we examined this limit by looking at the segment of the piecewise function where \( f(x) = x^2 \) and calculating \( \lim_{x \to 0^-} \frac{x^2}{x} = \lim_{x \to 0^-} x = 0 \).

The left-hand limit is essential when determining the differentiability and continuity of piecewise functions. It helps us understand the behavior of the function as we approach a point from the left side, showing if there's a smooth transition or a potential discontinuity.

Right-hand limit

The right-hand limit is analogous to the left-hand limit, but it considers the approach from the right-hand side of a point. This involves analyzing the function values that are just above the point.

• This limit can be expressed mathematically as \( \lim_{x \to a^+} f(x) \) for a given point \( a \).
• In our example function at \( x = 0 \), the right-hand limit is evaluated for the piece where \( f(x) = x^3 \), leading to \( \lim_{x \to 0^+} \frac{x^3}{x} = \lim_{x \to 0^+} x^2 = 0 \).

Ensuring the right-hand limit is equal to the left-hand limit is crucial for the function's differentiability and continuity at a specific point. If these limits exist and are equal at a point, it implies that the function is smooth and doesn't have a sharp corner or cusp there. This is why verifying both the left-hand and right-hand limits is a critical step in determining whether a function is differentiable at a given point.