|f(x)-f(y)| ≤ |x-y|ᵃ is called a-Hölder condition when a>0.
This condition implies that f(x) is Hölder continuous. Every Hölder continuous function is Uniformly Continuous.
∴ f:(a,b)→ℝ is Uniformly Continuous function.
If {xₙ} is any Cauchy sequence in (a,b), then {f(xₙ)} is a Cauchy sequence as f is uniformly continuous.
Consider xₙ = a+1/n, xₙ is Cauchy, hence f(xₙ) is also Cauchy and converges to some α, since ℝ is complete, and
Consider yₙ = b-1/n, yₙ is Cauchy, hence f(yₙ) is also Cauchy and converges to some β, since ℝ is complete.
Define g:[a,b]→ℝ as g(a) = α, g(b) = β, g(x) = f(x) when a<x<b.
It is not hard to prove the continuity of g at a and b using uniform continuity of f. So, the statement is ‘’true’’.