84 3.5 Uniform Continuity Definition 3.5.2 (Hölder continuity). Let Dbe a nonempty subset of R. A function f : D→Ris said to be Hölder continuous if there are constants ℓ ≥0 and α>0 such that | f (u)−f (v)| ≤ℓ|u−v|α for every u,v ∈D. The number α is called the Hölder exponent of the function. If α=1, then the function f is called Lipschitz continuous. Theorem 3.5.2 If a function f : D→Ris Hölder continuous, then it is uniformly continuous. Proof: Since f is Hölder continuous, there are constants ℓ ≥0 and α>0 such that | f (u)−f (v)| ≤ℓ|u−v|α for every u,v ∈D. If ℓ =0, then f is constant and, thus, uniformly continuous. Suppose next that ℓ >0. For any ε >0, let δ = ε ℓ 1/α. Then, whenever u,v ∈D, with|u−v| <δ we have | f (u)−f (v)| ≤ℓ|u−v|α < ℓδα =ε. The proof is now complete. □ Figure 3.3: The square root function. ■ Example 3.5.5 (a) Let D= [a,∞), where a >0. We will show that f (x) =√x is Lipschitz continuous on Dand, hence, uniformly continuous on this set. Givenu,v ∈D, we have | f (u)−f (v)| =| √ u− √v| = | u−v| √u+√v ≤ 1 2√a| u−v|, which shows f is Lipschitz continuous with ℓ =1/(2√a). (b) Let D= [0,∞). We will show that f (x) =√x is not Lipschitz continuous onD, but it is Hölder continuous on Dand, hence, f is also uniformly continuous on this set. Suppose by contradiction that f is Lipschitz continuous on D. Then there exists a constant ℓ >0 such that | f (u)−f (v)| =| √ u− √v| ≤ℓ|u−v| for every u,v ∈D. Considering v =0 and un =1/nfor every n∈N,we have 1 √n− 0 ≤ℓ 1 n− 0 .
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