Is the Function f(x) x3 / (1 - x2) Uniformly Continuous on the Real Numbers?

The question of whether a function is uniformly continuous is a fundamental discussion in mathematical analysis. In this article, we will explore the uniform continuity of the function f(x) x3 / (1 - x2) on the set of real numbers, ?. We will follow a systematic approach to determine its uniform continuity by examining its continuity, behavior at infinity, and the application of the definition of uniform continuity.

1. Continuity of f(x)

First, we need to check the continuity of the function f(x). The function is a rational function, and it is continuous everywhere on ? as long as the denominator 1 - x2 is not zero. Since 1 - x2 0 implies x ±1, the function f(x) is undefined at x ±1. However, excluding these points, the function is continuous on ?.

2. Behavior at Infinity

Next, we examine the behavior of f(x) as x approaches ±∞:

As ( x to ∞ ):h4>

[f(x) frac{x3}{1 - x2} frac{x3}{x2(1 - frac{1}{x2})} approx frac{x}{1 - 0} to ∞]

As ( x to -∞ ):h4>

[f(x) frac{x3}{1 - x2} frac{x3}{x2(1 - frac{1}{x2})} approx frac{x}{1 - 0} to -∞]

In both cases, the function does not approach a finite limit as x goes to ±∞. Instead, it diverges to ±∞.

3. Uniform Continuity

A key property of uniformly continuous functions is that they cannot diverge to infinity as x approaches ±∞. To show that f(x) is not uniformly continuous, we will use the definition of uniform continuity:

For every ε > 0, there exists a δ > 0 such that for all x, y ∈ ?, if |x - y|

Counterexample

We will provide a counterexample to demonstrate that f(x) is not uniformly continuous. Consider the sequences x_n n and y_n n - 1/n. As n approaches infinity:

x_n - y_n n - (n - 1/n) 1/n

As n → ∞, x_n - y_n → 0.

Now we compute f(x_n) and f(y_n) for large n:

f(x_n) f(n) frac{n3}{1 - n2} ≈ n

f(y_n) f(n - 1/n) frac{(n - 1/n)3}{1 - (n - 1/n)2} ≈ frac{n3 - 3n2 3n cdot frac{1}{n} - frac{1}{n2}}{1 - n2 2n cdot frac{1}{n} - frac{1}{n2}} ≈ frac{n3 - 3n2 3n - frac{1}{n}}{1 - n2 2 - frac{1}{n2}}

For large n, the terms involving 1/n and 1/n2 become negligible, so:

f(y_n) ≈ n3 - 3n2 3n

The difference between f(x_n) and f(y_n) can be significant:

|f(x_n) - f(y_n)| ≈ |n - (n - 3)| 3

Thus, for any fixed ε , no matter how small we choose δ, there will always be values of x_n and y_n such that |x_n - y_n| δ but |f(x_n) - f(y_n)| ≥ 3. Therefore, the condition |f(x) - f(y)| cannot be satisfied, and f(x) is not uniformly continuous on ?.

Conclusion

Based on the analysis, we conclude that the function f(x) x3 / (1 - x2) is not uniformly continuous on the real numbers ?.