1.7 Precise definition of a limit (極限之嚴格定義)

the limit of f(x) as x approaches a is L

$$
\lim_{x \to a}f(x)=L
$$

if for every number ε>0 there is a corresponding number δ>0 such that

$$
if \quad 0<|x-a|<δ \quad then \quad |f(x)-L|<ε
$$

(∀ε>0, ∃ δ>0 s.t. 0<|x-a|<δ implies |f(x)−L|<ε)

• Infinite Limits (無窮極限之嚴格定義)

the limit of f(x) as x approaches a is ∞

$$
\lim_{x \to a}f(x)=∞
$$

means that for every position number M there is a position number δ such that

$$
if \quad 0<|x-a|<δ \quad then \quad f(x)>M
$$

(∀M>0, ∃ δ>0 s.t. 0<|x-a|<δ implies f(x)>M)

寫程式的門檻不算太高[name=白惠明]

1.8 Continuity 連續性

A function f is continuous at a number a if

$$
\lim_{x \to a} f(x) = f(a)
$$

• 複合函數之極限性質

$$
\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x))
$$

• 中值定理

若f為連續且f(a)≠f(b),則f由f(a)移至f(b)必經過中間所有數值