Problem 13 - Basic Analysis: Introduction to Real Analysis

Question

Suppose \(f:(-1,1) \rightarrow \mathbb{R}\) is a function such that \(f(x)=x h(x)\) for a bounded function \(h .\) a) Show that \(g(x):=(f(x))^{2}\) is differentiable at the origin and \(g^{\prime}(0)=0\). b) Find an example of a continuous function \(f:(-1,1) \rightarrow \mathbb{R}\) with \(f(0)=0,\) but such that \(g(x):=(f(x))^{2}\) is not differentiable at the origin.

Step-by-Step Solution

Answer

(a) Differentiable at origin with \(g'(0) = 0\). (b) Example: \(f(x) = x^2 \sin(\frac{1}{x})\) for \(x \neq 0\), \(f(0) = 0\).

Step 1: Express the Function

Substitute the given function into the expression for \(g(x)\):\[ g(x) = (f(x))^2 = (xh(x))^2. \] Simplify it to get the relation in terms of \(x\):\[ g(x) = x^2 (h(x))^2. \]

Step 2: Calculate the Derivative of g(x)

Use the derivative rules, particularly the power rule and the constant multiple rule. Differentiate \(g(x) = x^2 (h(x))^2\):\[ g'(x) = 2x (h(x))^2 + x^2 \cdot 2h(x)h'(x). \] Evaluate at \(x = 0\):\[ g'(0) = 2 \cdot 0 \cdot (h(0))^2 + 0^2 \cdot 2h(0)h'(0) = 0. \] Thus, \(g'(0) = 0\).

Step 3: Example of Non-differentiable Function at Origin

Choose \(f(x)\) as \(f(x) = x^2 \sin \left( \frac{1}{x} \right)\) for \(x eq 0\), and \(f(0) = 0\). This \(f\) is continuous and satisfies \(f(0) = 0\). For \(g(x) = (f(x))^2\), calculate \(g(x)\) as:\[ g(x) = (x^2 \sin \left( \frac{1}{x} \right))^2 = x^4 \sin^2 \left( \frac{1}{x} \right). \]The derivative \(g'(x)\) will include terms derived from \(x^4\) and \(\sin^2\left( \frac{1}{x} \right)\) whose behavior, especially the oscillating nature as \(x \to 0\), may lead to non-differentiability.

Key Concepts

Differentiability

Differentiability is a core concept in calculus that deals with the behavior of functions and their smoothness. A function is differentiable at a point if it has a derivative at that point. This essentially means that there is a defined slope or tangent at that point, indicating smoothness.

• For a function to be differentiable, it must first be continuous.
• Differentiability implies local linearity, where the function resembles a straight line in the immediate vicinity of the point.

In the exercise, we established that for the function \(g(x) = x^2 (h(x))^2\), the derivative exists at the origin, showing the function is differentiable there.
Differentiability excludes corners or cusps, which means that a graph cannot abruptly change direction at a differentiable point. A classic example is the function \(f(x) = |x|\), which is not differentiable at \(x = 0\) due to the abrupt cusp.

Continuous Functions

Continuous functions are functions without any breaks, jumps, or holes in their graphs. Continuity ensures smooth progression without sudden changes.

• Mathematically, a function \(f(x)\) is continuous at a point \(x = a\) if \(\lim_{x \to a} f(x) = f(a)\).
• A function may be continuous on an interval if it is continuous at every point in that interval.

In our problem, we examined \(f:(-1,1) \to \mathbb{R}\), a continuous function defined on the interval \((-1,1)\) such that \(f(0) = 0\).
Continuity paved the path for us to explore differentiability; however, it does not guarantee differentiability. A function can be continuous yet not differentiable, as shown with \(f(x) = x^2 \sin \left( \frac{1}{x} \right)\) for \(x eq 0\) and \(f(0) = 0\).

Derivative Calculation

The derivative calculation reveals the function's rate of change at any given point. It's the cornerstone for understanding dynamic behavior.

• Derivatives can be found using various rules, including the power rule, product rule, and chain rule.
• The power rule helps when differentiating terms like \(x^n\), with the derivative being \(nx^{n-1}\).

In the step-by-step solution, we calculated the derivative \(g'(x)\) for \(g(x) = x^2 (h(x))^2\) at \(x = 0\). The calculation showed that \(g'(0) = 0\), verifying differentiability at the origin.
Exploring derivatives involves not just direct calculation but also understanding the implications of these values. For example, a zero derivative indicates a potential local maximum or minimum, crucial in graph analysis and optimization. Understanding these can greatly aid in solving complex problems in real analysis.