64. Suppose that f is a continuous real-valued function defined on the closed interval [0,1]. Which of the following must be true?
- I. There is a constant C > 0 such that | f(x) - f(y) | ≤ C for all x and y in [0,1].
- II. There is a constant D > 0 such that | f(x) - f(y) | ≤ 1 for all x and y in [0,1] that satisfy |x-y| ≤ D.
- III. There is a constant E > 0 such that | f(x) - f(y) | ≤ E |x-y| for all x and y in [0,1].
(A) I only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III
The definition of continuity: $f$ is continuous at $x = a$ if for all $\epsilon > 0$, there exists $\delta > 0$ such that $0 < |x-a| < \delta $ implies $|f(x) - f(a)| < \epsilon$.
Since $f$ is continuous on a closed interval $[0,1]$, we see that $f$ is also uniformly continuous on $[0,1]$.
The definition of uniformly continuous: $f$ is uniformly continuous on $E = [a,b]$ if given $\epsilon > 0$ and $x, y \in E$, there exists $\delta > 0$ such that $|f(x) - f(y)| < \epsilon$ whenever $|x-y| < \delta$.
I is true, with $C = \epsilon > 0$.
II must be true, with $\delta = D > 0$ and $\epsilon = 1$.
III is actually false. Yes, if we have this, then it implies $f$ is uniformly continuous. But what we have is actually the opposite. And we know the converse of a true statement is not necessarily still true. As for a counter example, take $f(x) = x^n$ for really large $n$.
The answer is C.