Problem 1 - A Primer of Real Analysis

Question

Show that if \(c \in \mathbb{R}\) and \(f(x)=c\) for all \(x \in \mathbb{R},\) then \(f^{\prime}(x)=0\) for all \(x \in \mathbb{R}\).

Step-by-Step Solution

Answer

For a constant function \(f(x) = c\), the derivative \(f'(x) = 0\) for all \(x \in \mathbb{R}\).

Step 1: Understand the function and its continuity

The function given is constant, i.e., \(f(x) = c\) for all \(x \in \mathbb{R}\). A constant function doesn't change its value regardless of the input \(x\).

Step 2: Recall the definition of a derivative

The derivative of a function \(f\) at a point \(x\) is defined as \(f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}\). This expresses the rate of change or the slope of the function at that point.

Step 3: Apply the derivative definition to the given function

For the function \(f(x) = c\), we have \(f(x+h) = c\) as well since \(f\) is constant. Substitute these into the derivative formula: \[f'(x) = \lim_{{h \to 0}} \frac{c - c}{h} = \lim_{{h \to 0}} \frac{0}{h} = 0.\]

Step 4: Confirm the result for all \(x \in \mathbb{R}\)

Since the function is constant and independent of \(x\), the derivative calculation is valid for any value of \(x\). Therefore, \(f'(x) = 0\) for all \(x \in \mathbb{R}\).

Key Concepts

Constant Function

A constant function is one of the simplest forms of a function. It is expressed as \(f(x) = c\), where \(c\) is a constant real number, and \(x\) can be any real number from the set of all real numbers \(\mathbb{R}\).

In a constant function, no matter what value you substitute for \(x\), the output of the function will always be the same fixed value, \(c\). This means that the graph of a constant function is a horizontal line.

• The slope of a horizontal line is zero.
• This implies no change in the function's value as \(x\) varies.

Understanding the concept of a constant function is essential as it simplifies the differentiation process, which we'll explore next.

Derivative Definition

The derivative represents a fundamental concept in calculus that describes how a function's value changes as its input changes.

Mathematically, the derivative \(f'(x)\) of a function \(f(x)\) at a point \(x\) is defined as:
\[f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}\]

This expression calculates the average rate of change of the function over an interval \(h\), and then takes the limit as \(h\) approaches zero. This results in the exact rate of change at the point \(x\).

• The derivative tells you the slope of the tangent line to the curve at any given point.
• In the case of a constant function, it simplifies to finding how flat the function is everywhere.

Understanding how to apply this formula helps you find the derivative of any smooth function.

Rate of Change

The rate of change in the context of differentiable functions is essentially what the derivative captures. It's the speed or level of change occurring in a function's value relative to changes in its input.

For a constant function like \(f(x) = c\), no matter what happens to \(x\), the output, \(c\), stays the same. Therefore, the rate of change is zero.

• No change means that the function's graph is perfectly horizontal.
• The slope of this horizontal line, and thus its rate of change, is zero.

Hence, the derivative of a constant function is always zero, confirming that such functions do not change.

Real Numbers

Real numbers, denoted by \(\mathbb{R}\), encompass every number that can exist on the number line including rational and irrational numbers.

When working with constant functions such as \(f(x) = c\), \(x\) can belong to any real number, meaning the function is defined and constant across all of \(\mathbb{R}\).

• Real numbers can be positive, negative, or zero.
• This gives the constant function a broad scope of applicability.

Since a constant function is uniformly \(c\) everywhere in its domain, its derivative, which calculates changes, remains zero across the entire set of real numbers. This characteristic underscores the uniformity of constant functions.