91Ó°ÊÓ

A **constant function** is a very simple but crucial concept in calculus and real analysis. A function \( f(x) \) is considered constant if, for any two points \( x \) and \( y \) in its domain, the function always produces the same value. Mathematically, this can be expressed as \( f(x) = f(y) \).
This means the function graph is a horizontal line.

These functions are important because they highlight the notion of rate of change.
In a constant function, the rate of change is zero, indicating that no matter how you move along the x-axis, the y-value remains unchanged.

**Real analysis** is a branch of mathematics that deals with real numbers and real-valued functions. In real analysis, we rigorously study various properties and behaviors of functions.
It involves concepts like sequences, limits, continuity, derivatives, and integrals.
This discipline provides a foundation for much of modern mathematics.

The problem at hand, concerning whether a function is constant, is rooted in real analysis.
The definition given in the problem, \( |f(x) - f(y)| \, \leq \, (x - y)^2 \), is an inequality that tells us about the behavior of \( f \) over the real numbers \( \mathbb{R} \).
This definition is part of understanding how functions behave more generally.

The **limit process** is an essential concept in calculus, which involves finding the value that a function approaches as the input approaches some point.
In our exercise, we used the limit process to analyze the behavior of \( f \) at extremely small intervals.

We considered \( y = x + h \) where \( h \) is very small.
This let us write the given condition as \( |f(x) - f(x + h)| \, \leq \, h^2 \).
By dividing both sides of this inequality by \( h \), we get \( |\frac{f(x) - f(x + h)}{h}| \, \leq \, h \). As \( h \) approaches zero, the term \( \frac{f(x) - f(x + h)}{h} \) needs to approach zero.
This is essentially finding the derivative and establishing that it must be zero, indicating no change in \( f \) and proving that \( f \) is constant.

The **derivative** measures the rate at which a function changes as its input changes.
Mathematically, it is defined as:

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

In our given problem,
we started with \( |f(x) - f(x + h)| \, \leq \, h^2 \) and manipulated this inequality to find \( \frac{f(x) - f(x+h)}{h} \, \leq \, h \).

Taking the limit as \( h \) approaches zero,
the left-hand side, which is the definition of the derivative of \( f \) at \( x \), also approaches zero.
This implies \( f'(x) = 0 \).

If the derivative is zero for all \( x \), the function \( f \) must be constant across its domain.

Inequalities play a significant role in calculus by providing bounds and constraints on functions.
In our given exercise, the inequality \( |f(x) - f(y)| \, \leq \, (x - y)^2 \) helped to analyze the behavior of the function \( f \).

Understanding how small changes in \( x \) and \( y \) affect \( f(x) \) using inequalities enabled us to infer properties about the function.

These kinds of inequalities are essential for:

• Proving continuity
• Finding Lipschitz conditions
• Analyzing differentiability

Inequalities can provide critical insights into the stability and uniqueness of solutions in calculus.