Introduction to Analysis: Fall 2004
Practice problems V1
MTH 4101/5101 11/22/2004
1. Determine whether the given functions are continuous. Explain clearly.
(a) f(x)=|x|
x,where x=±1
n,n∈IN
(b) f(x)=(x sin(1
x)ifx6=0
0ifx=0
(c) f(x)=(x2sin(1
x)ifx6=0
0ifx=0
(d) f(x)=(e(−1
x2)if x6=0
0ifx=0
2. Determine if the functions given below are continuous or not at the given
points.
(i) f(x)=(xif x=1
nand n∈Z\{0}
1−xotherwise
(ii) f(x)=(xif xis rational
1−xif xis irrational
3. Suppose that a function fis continuous on (a, b) and f(r)=cfor all rational
rin (a, b) and cis some fixed real constant. Prove that f(x)=cfor all
x∈(a, b).
4. Give examples of the following type of fucntions:
(i) function fdefined on IR but not continuous at any point of IR
(ii) function fdefined on IR but continuous at exactly at one point of IR
(iii) function fdefined on [a, b] but continuous only at denumerably many
points of [a, b].
5. For each of the given functions, locate and classify all the points of
discontinuity.
(i) f(x)=|x|
x,where x=±1
n,n∈IN
(ii) f(x)=bxc+x
2,x∈(−3
2,1]
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