Problem 54 - Calculus Single Variable

Question

Find a continuous function \(f\) on \(\mathbb{R}\) that is differentiable on \((-\infty, 0) \cup(0, \infty)\) such that \(f(0)=0\) and \(f^{\prime}(x)=H(x)\) for \(x \neq 0\). Here \(H\) is the Heaviside function: $$ H(x)=\left\\{\begin{array}{lll} 0 & \text { if } & x<0 \\ 1 & \text { if } & x>0 \end{array}\right. $$.

Step-by-Step Solution

Answer

The function is \( f(x) = 0 \) for \( x < 0 \) and \( f(x) = x \) for \( x \geq 0 \).

Step 1: Understand the Heaviside Function

The Heaviside function, denoted by \( H(x) \), is defined as 0 for \( x < 0 \) and 1 for \( x > 0 \). It is not defined or discontinuous at \( x = 0 \). This function represents a step change at the origin.

Step 2: Analyze the Requirements for Function \( f \)

We need a continuous function \( f(x) \) which is differentiable everywhere except at \( x = 0 \) and has the property that \( f'(x) = H(x) \) for all \( x eq 0 \). Additionally, \( f(0) = 0 \).

Step 3: Determine \( f(x) \) for \( x < 0 \)

For \( x < 0 \), \( f'(x) = 0 \) since \( H(x) = 0 \). Integrating this, we get \( f(x) = C_1 \). Since \( f(0) = 0 \), and for continuity at 0, \( C_1 = 0 \). Thus, \( f(x) = 0 \) for all \( x < 0 \).

Step 4: Determine \( f(x) \) for \( x > 0 \)

For \( x > 0 \), \( f'(x) = 1 \) since \( H(x) = 1 \). Integrating \( 1 \) in the region \( x > 0 \), we get \( f(x) = x + C_2 \). For continuity at \( x = 0 \), if \( f(0) = 0 \), we must have \( f(0^+) = 0 \), thus \( C_2 = 0 \). So, \( f(x) = x \) for all \( x > 0 \).

Step 5: Construct the Full Function

The continuous function \( f(x) \) that satisfies all the conditions is therefore piecewise: \[f(x) = \begin{cases} 0 & \text{if } x < 0 \x & \text{if } x \geq 0\end{cases}\] This function is continuous everywhere, differentiable everywhere except \( x = 0 \), and satisfies \( f'(x) = H(x) \) for \( x eq 0 \).

Key Concepts

Piecewise Functions

In mathematics, a piecewise function is a type of function that is defined by multiple sub-functions, each corresponding to a specific part of the domain. These functions are often used when a single formula cannot define the function over its entire domain.

A piecewise function looks like a series of pieces connected in different parts of the graph. For instance, the function we discussed in the original problem is defined piecewise. It is:

• \( f(x) = 0 ext{ for } x < 0 \, \)
• \( f(x) = x ext{ for } x \geq 0 \, \)

This function is continuous because each piece smoothly connects to the next, ensuring there are no breaks or gaps in the graph.

In real-world situations, piecewise functions can model anything from tax brackets to the changing speed of a car. They are invaluable in situations where a single rule does not apply uniformly.

Heaviside Function

The Heaviside function, also known as the unit step function, is a fundamental discontinuous function in control theory and signal processing. It transitions between two values abruptly:

• For any point where \( x < 0 \), \( H(x) = 0 \).
• For any point where \( x > 0 \), \( H(x) = 1 \).
• The function is undefined or left out at \( x = 0 \), causing a discontinuity.

The Heaviside function is often used in mathematical models to depict switch-like behaviors. It represents an instantaneous shift, useful in physics to model force applications, electrical circuits, and more.

When used as a derivative in calculus, as seen in the original exercise, the Heaviside function indicates a step change in the graph of a function. This makes it a valuable tool in signals and systems, enabling the modeling of processes that have a clear cut on and off behavior.

Differentiability

Differentiability is a concept in calculus that describes if a function has a derivative at each point in its domain. Simply put, if you can find a tangent line to the function's graph at any specific point, the function is differentiable at that point.

In the context of the exercise, we aim to find a function that is differentiable on \((-infty, 0) \cup (0, infty)\) except precisely at \( x = 0 \).

• Differentiability ensures that the function behaves predictably and smoothly except at certain points.
• Our original problem function is:\[ f(x) = \begin{cases} 0 & \text{if } x < 0 \ x & \text{if } x \geq 0 \end{cases} \]

This specific function is differentiable everywhere except at \( x = 0 \)because of the discontinuity created by the Heaviside function. The derivative exists on either side of the point but not at the point itself, reflecting that smooth transition occurs everywhere else in the defined domain.