Fractional differentiability introduces a fascinating twist to calculus, extending the idea of derivatives to non-integer orders. We’ll explore a family of functions that are fractionally differentiable up to a certain order but not beyond. Understanding fractional derivatives allows us to analyze functions with nuanced behaviors. By examining power functions and absolute values, we can construct functions that exhibit specific fractional differentiability properties. Let’s see how these concepts come together.

---

---

Welcome to an exploration of fractional calculus! Today, we’re diving into a fascinating question: can we find a family of functions, denoted as ##f_a(x)##, that are fractionally differentiable to a certain order ##a## but not beyond? This concept extends the traditional integer-order derivatives to real numbers, opening up new possibilities in mathematical analysis. Let’s explore this intriguing problem and its solution.

Understanding Fractional Derivatives

Fractional derivatives generalize the concept of integer-order derivatives to non-integer orders. The ##a##-fractional derivative of a function ##f(x)##, where ##a## is a real number, is denoted as ##D^a f(x)##. These derivatives can be defined using various approaches, such as the Riemann-Liouville or Caputo definitions. The key idea is to extend the familiar rules of differentiation to fractional orders.

One common example is the power function ##x^a##, which is fractionally differentiable for ##x > 0##. The fractional derivative of ##x^a## is proportional to ##x^{a-b}##, where ##b## is the order of the fractional derivative. However, differentiability issues arise when considering functions at points where their behavior is not smooth, such as at ##x = 0##.

Constructing a Family of Functions

Fractional Differentiability and Power Functions

Consider the family of functions ##f_a(x) = x^a##. For ##x > 0##, the ##b##-fractional derivative of ##f_a(x)## exists if ##b \leq a##. Specifically, the ##b##-fractional derivative is proportional to ##x^{a-b}##. However, if ##b > a##, the fractional derivative typically blows up at the origin, meaning it does not exist. This behavior provides a starting point for constructing the desired family of functions.

To illustrate, let’s compute the first derivative of ##f(x) = x^2##, which is ##f'(x) = 2x##. Now, if we consider a fractional derivative of order 1.5 (i.e., ##b = 1.5##), the result would be proportional to ##x^{2-1.5} = x^{0.5}##, which is well-defined for ##x > 0##. However, the second derivative would be a constant, and further fractional derivatives may not exist in a traditional sense.

Using Absolute Value for Non-Differentiability

To create functions that are fractionally differentiable up to a certain order but not beyond, consider the fractional anti-derivatives of ##|x|##. The absolute value function ##|x|## is defined as ##x## for ##x > 0## and ##-x## for ##x \leq 0##. This function is continuous but not differentiable at ##x = 0##. Taking fractional anti-derivatives can smooth out this non-differentiability to a certain extent.

The ##k##th anti-derivative of ##x## is given by ##\frac{x^{k+1}}{(k+1)!}##. Therefore, we can construct a family of functions by taking fractional anti-derivatives of ##|x|##. For example, the first anti-derivative of ##|x|## is ##\frac{x^2}{2}## for ##x > 0## and ##-\frac{x^2}{2}## for ##x < 0##. These functions are easier to find because we can treat ##|x|## as piecewise functions and apply standard integration formulas.

Final Solution

The family of functions ##f_a(x)## can be constructed by taking the ##a##-fractional anti-derivatives of ##|x|##. These functions are fractionally differentiable up to order ##a##, but their derivatives of order greater than ##a## do not exist in a well-defined sense. This construction leverages the non-differentiability of ##|x|## at the origin to create functions with specific fractional differentiability properties.

In summary, the function ##x^a## is fractionally differentiable at ##x=0## with a ##b##-fractional derivative proportional to ##x^{a-b}## for ##b \leq a##. However, if ##b > a##, the fractional derivative blows up at the origin. Therefore, by considering the ##a##-fractional anti-derivatives of ##|x|##, we can create a collection of functions that have ##a##-fractional derivatives but not ##b##-fractional derivatives for ##b > a##.

Similar Problems and Quick Solutions

Problem 1: Find the derivative of ##f(x) = x^3##

Solution: ##f'(x) = 3x^2##

Problem 2: Find the integral of ##f(x) = x^4##

Solution: ##\int f(x) dx = \frac{x^5}{5} + C##

Problem 3: Determine if ##f(x) = |x^2|## is differentiable at ##x = 0##

Solution: Yes, ##f(x) = x^2##, so ##f'(x) = 2x## and ##f'(0) = 0##.

Problem 4: Find the first anti-derivative of ##f(x) = x##

Solution: ##F(x) = \frac{x^2}{2} + C##

Problem 5: Determine the derivative of ##f(x) = sin(x)##

Solution: ##f'(x) = cos(x)##

---

---