Proving the Uniqueness of a Continuous Function in Relation to the Arctangent

Introduction to the Problem

Consider the equation:

(y arctan (xy))

Our objective is to determine if there exists a continuous function (f(y)) such that (y arctan (xf(y))). If such a function exists, how can we prove its uniqueness?

Understanding the Concept of an Arctangent Function

The arctangent function, (arctan(x)), is the inverse function of the tangent function. However, since the tangent function is not one-to-one over its entire domain, the arctangent function is defined as

(arctan(x) theta)

such that

(-frac{pi}{2}

and

(tan(theta) x).

Deriving the Unique Function

From the equation given, we have:

(y arctan (xy))

Since the arctangent function is the inverse of the tangent function, we can rewrite this as:

(tan(y) xy)

To solve for (x), we get:

(x frac{tan(y)}{y})

This shows that for any given (y) (excluding (y 0)), there is a unique value of (x).

Proving Uniqueness and Continuity

To prove the uniqueness of the function (f(y) frac{tan(y)}{y}), we must consider the properties of the arctangent function and the function itself:

Uniqueness: For any nonzero (y), the function (f(y) frac{tan(y)}{y}) is uniquely defined. This is because the arctangent function is strictly increasing and the tangent function is strictly increasing in the interval where we are considering it. Therefore, the function (x frac{tan(y)}{y}) is uniquely determined for each (y), apart from the origin where division by zero is undefined.

Continuity: The function (f(y)) is continuous for all (y eq 0). We can analyze the behavior of (frac{tan(y)}{y}) as (y) approaches zero to establish its continuity at the origin. Since (lim_{y to 0} frac{tan(y)}{y} 1), the function is continuous at (y 0) when defined as (f(0) 1).

Conclusion: Existence and Uniqueness of a Continuous Function

We have shown that there exists a unique and continuous function (f(y) frac{tan(y)}{y}) such that:

(y arctan (xf(y)))

for all (y eq 0). This function is continuous for all (y) and matches the given condition. Thus, we can confidently say that (f(y)) is the unique continuous function satisfying the given equation.

Keywords: continuous function, arctangent, inverse function