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Lebl-Basic-Analysis-I
· Problem 3.1.14
Continuous Functions
Mathematics · limits-of-functions
Problem
(Challenging) Show that the hypothesis that
g
(
c
2
)
=
L
g(c_2) = L
g
(
c
2
)
=
L
in Exercise 3.1.9 is necessary. That is, find
f
f
f
and
g
g
g
such that
f
(
x
)
→
c
2
f(x) \to c_2
f
(
x
)
→
c
2
as
x
→
c
1
x \to c_1
x
→
c
1
and
g
(
y
)
→
L
g(y) \to L
g
(
y
)
→
L
as
y
→
c
2
y \to c_2
y
→
c
2
, but
g
(
f
(
x
)
)
g\bigl(f(x)\bigr)
g
(
f
(
x
)
)
does not go to
L
L
L
as
x
→
c
1
x \to c_1
x
→
c
1
.
Related theory
Limits And Continuity Of Real Valued Functions
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