高微問題求解 小弟第一次發問有錯誤請見諒?

Question

1.

Let f:R→R be a function defined on R. Please give a definition for f(X) to be uniformly continuous.

2.

Let f:R→R be a function defined on R .Please use ε-δ language to define what it mean for the function f(X) to be continuous at X=1.

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Answer 1

最佳解答

Let f:R→R be a function defined on R. The function f is uniformly continuous over R

if for any ε > 0 there exists a δ > 0 such that whenever |x-y| < δ then |f(x)-f(y)| < ε.

在不是考慮整個數線, 而是一個實數的子集的情形:

A function f is said to be uniformly continuous over a subset D of R if:

for any ε > 0 there exists a δ > 0 such that

whenever |x-y| < δ, x, y in D, then |f(x)-f(y)| < ε.

至於在一點 x=a 連績 (你可代之以 x=1):

Let f:R→R be a function defined on R. The function f is said to be continuous at x=a

if for any ε > 0 there exists a δ > 0 such that whenever |x-a| < δ then |f(x)-f(a)| < ε.

而在一個實數子集逐點連續 (point-wise continuous):

A function f is said to be continuous over a subset D of R if

for any x in D, any ε > 0 there exists a δ > 0 such that

whenever |y-x| < δ then |f(y)-f(x)| < ε.

此定義與 uniformly continuous 不同在於:

point-wise continuous 是先決定一點再決定 δ > 0, 然後考慮 ,

因此 δ 可以與 x 有關, 也就是說不同 x 用不同 δ > 0; 而

uniformly continuous 則是在決定了 ε > 0 後就決定了 δ ,

然後要適用於所有的 x (及 y) in D, δ 不能隨 x 而調整.