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SECTION 2.5 CONTINUITY
❙❙❙❙
125
Now let’s see how to detect discontinuities when a function is defined by a formula.
EXAMPLE 2 Where are each of the following functions discontinuous? Resources / Module 2 / Continuity / Problems and Tests
x2 x 2 (a) f x x2 (c) f x
(b) f x
x2 x 2 x2 1
if x 2
1 x2 1
if x 0 if x 0
(d) f x x
if x 2
SOLUTION
(a) Notice that f 2 is not defined, so f is discontinuous at 2. Later we’ll see why f is continuous at all other numbers. (b) Here f 0 1 is defined but lim f x lim
xl0
xl0
1 x2
does not exist. (See Example 8 in Section 2.2.) So f is discontinuous at 0. (c) Here f 2 1 is defined and lim f x lim x l2
x l2
x2 x 2 x 2x 1 lim lim x 1 3 x l2 x l2 x2 x2
exists. But lim f x f 2 x l2
so f is not continuous at 2. (d) The greatest integer function f x x has discontinuities at all of the integers because lim x ln x does not exist if n is an integer. (See Example 10 and Exercise 49 in Section 2.3.) Aşağıdaki grafikler yukarıdaki örnekteki fonksiyonların grafikleridir.Her bir durumda grafiği kalemi kaldırmadan çizemiyoruz, çünkü bu grafiklerde bir boşluk veya bir kesiklik veya bir sıçrama ile karşılaşıyoruz. Burada (a) 'daki ve (c) 'deki süreksizliklere kaldırılabilir süreksizlik (f yi x=2 noktasında tekrar tanımlayarak bu süreksizliği ortadan kaldırabiliriz), (b) 'deki süreksizliğe sonsuz süreksizlik ve (d) 'deki süreksizliğe de sıçramalı süreksizlik (fonksiyon bir değerden diğerine atlıyor) diyeceğiz.
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SECTION 2.5 CONTINUITY
❙❙❙❙
125
Now let’s see how to detect discontinuities when a function is defined by a formula.
EXAMPLE 2 Where are each of the following functions discontinuous? Resources / Module 2 / Continuity / Problems and Tests
x2 x 2 (a) f x x2 (c) f x
(b) f x
x2 x 2 x2 1
if x 2
1 x2 1
if x 0 if x 0
(d) f x x
if x 2
SOLUTION
(a) Notice that f 2 is not defined, so f is discontinuous at 2. Later we’ll see why f is continuous at all other numbers. (b) Here f 0 1 is defined but lim f x lim
xl0
xl0
1 x2
does not exist. (See Example 8 in Section 2.2.) So f is discontinuous at 0. (c) Here f 2 1 is defined and lim f x lim x l2
x l2
x2 x 2 x 2x 1 lim lim x 1 3 x l2 x l2 x2 x2
exists. But lim f x f 2 x l2
so f is not continuous at 2. (d) The greatest integer function f x x has discontinuities at all of the integers because lim x ln x does not exist if n is an integer. (See Example 10 and Exercise 49 in Section 2.3.) Aşağıdaki grafikler yukarıdaki örnekteki fonksiyonların grafikleridir.Her bir durumda grafiği kalemi kaldırmadan çizemiyoruz, çünkü bu grafiklerde bir boşluk veya bir kesiklik veya bir sıçrama ile karşılaşıyoruz. Burada (a) 'daki ve (c) 'deki süreksizliklere kaldırılabilir süreksizlik (f yi x=2 noktasında tekrar tanımlayarak bu süreksizliği ortadan kaldırabiliriz), (b) 'deki süreksizliğe sonsuz süreksizlik ve (d) 'deki süreksizliğe de sıçramalı süreksizlik (fonksiyon bir değerden diğerine atlıyor) diyeceğiz.
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