連續函數的定義
Author: 藍邦偉  老師

$ O $

$ 連續函數的定義 $

$ x=c $

$ y=f(x) $

$(c,0) $

$ 函數f(x)滿足下列條件$
$(1)\;f(c)有定義$
$(2)\;\lim_{ x \to c }f(x)存在$
$(3)\;\lim_{ x \to c }f(x)=f(c)$
$則稱函數f(x)在x=c連續 $
$若在此定義中任何條件不成立$
$則稱\;f\;在\;c\;為不連續$
$若f在開區間[a,b]中所有的點$
$都連續，且\lim_{ x \to a^+ }f(x)=f(a)$
$\lim_{ x \to b^- }f(x)=f(b)，則稱f(x)在$
$[a,b]上連續.\;若函數f(x)在定$
$義域上每一點都連續，則稱$
$f(x)\;為連續函數.$

$∵函數f(x)在x=c沒有定義$
$∴  f(x)在x=c不連續$

$ ∵\lim_{ x \to c^+ }f(x)≠\lim_{ x \to c^- } f(x)$
$∴f(x)在x=c不連續$

$ ∵\lim_{ x \to c }f(x)≠f(c) $
$∴f(x)在x=c不連續$

$ ∵\lim_{ x \to c }f(x)不存在 $
$∴y=f(x)在x=c處不連續$

u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
Q_2: 0.7 * Corner[1] + 0.3 * Corner[2]
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
O = (0, 0)
Q_3: (x(Q_2), y(Corner[4]))
poly1, q_1, q_2, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
n = 0 (slider from 0 to 4, step 1)
A_1 = (-2.2211958601064725, 1.7796839316318158)
A_2 = (1.7894033688035689, 4.491612934037652)
A_3 = (4.367645730245739, 5.0645556810248005)
f_1 = Polynomial({A_1, A_2, A_3})
A = Point(f_1)
c_1 = Circle(A, 0.08)
c_2 = Circle(A, 0.12)
Z_1 = Point(f_1)
Z_3 = Point(f_1)
Z_2 = Point(f_1)
B_4 = Point(f_1)
B_5 = Point(f_1)
B_6 = Point(f_1)
B_2: Z_3 - (0, 3)
B_1: Z_1 - (0, 3)
B_3: Z_2 - (0, 3)
f_{2R} = Function(Polynomial[{B_4, B_5, B_6}], x(A) + 0.1, x(Corner[2]))
A_a: A - (0, 3)
c_4 = Circle(A_a, 0.12)
f_{2L} = Function(Polynomial[{B_1, B_2, B_3}], x(Q_2), x(A) - 0.1)
A_b: A - (0, 1.5)
a: x = x(A)
f_3(x) = 1 / (x - x(A))^(2)
b_1 = Segment((x(A), 0), (x(A), y(A) - 0.1))
t = 1.8519999999999999 (slider from x(Q_2) + 1 to x(Corner[2]) - 1, step 0.01)
B = (-6.382843631404034, 4.442999610050864)
text4: "$f(x)=" + (LaTeX[f_3]) + " $"
C = Point(a)
c_3 = Circle(A_a, 0.1)