※ 引述《pobm (妳會更快樂嗎?)》之銘言：
: Show that f(x)=√x is unif conti on [0,∞)
: 從圖上看還滿明顯的
: 我想說因為在compact[0,1] conti 所以unif 在[0,1]
硬著頭皮，寫出第一句：
Proof. Let ε > 0.
Since f is continuous, f restricted to the compact
interval [0,1] is uniformly continuous, and so
there exists η > 0 such that
|f(x) - f(y)| < ε
whenever x and y are in [0,1] and |x - y| < η.
: 其他變化量地方沒那麼大所以用同[0,1]的δ就可以
（★ 選擇 δ > 0 -- 待填）
Now suppose x and y are in [0,∞) and |x - y| < δ.
I will show that |f(x) - f(y)| < ε.
Case 1: If x and y are in [0,1], then
|f(x) - f(y)| < η.
（筆記：選擇 δ ≦ η。）
Case 2: Otherwise, we have √x + √y > 1, and so
|f(x) - f(y)| = |√x - √y| (√x + √y) / (√x + √y)
= |x - y| / (√x + √y)
< |x - y|
< δ.
（筆記：選擇 δ ≦ ε。）
（回去補充★： Let δ = min {η, ε}.）
In both cases, we have |f(x) - f(y)| < ε,
which is what we set out to prove. ::
: 不過不知道怎麼寫@@
: 謝謝
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