Question

Let f(x) = 1/x on [a,[infinity]) where a > 0. is f uniformly continuous?

Answer 1

Yes, the function f(x) = 1/x is uniformly continuous on [a, ∞) where a > 0.

Step-by-step explanation:

Uniform Continuity: A function f(x) is uniformly continuous if for any given value of ε > 0, there exists a δ > 0 such that if |x - y| < δ, then |f(x) - f(y)| < ε.

To determine if f(x) = 1/x is uniformly continuous on [a, ∞), we need to show that for any ε > 0, there exists a δ > 0 such that if |x - y| < δ, then |1/x - 1/y| < ε.

Let's suppose a > 0 and choose δ = aε/(a+1). Then, for any |x - y| < δ, we have:

|1/x - 1/y| = |(y - x)/(xy)| ≤ |x - y|/(a^2) < δ/(a^2) = aε/(a+1) / (a^2) = ε / (a+1).

Since ε / (a+1) < ε, we can conclude that f(x) = 1/x is uniformly continuous on [a, ∞) for any a > 0.