Check whether the set of integers is countable or not.
The set of integers ( \mathbb{Z} = { \dots, -3, -2, -1, 0, 1, 2, 3, \dots } ) can be shown to be countable by explicitly constructing a one-to-one correspondence ________ ________ _____ _________ ___ __________ _________ _________ ___ ________.
______ ___ ___ _______ ________.
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______ ________ ________ ______ ________ ______ _______ __________ ________.
_____ _________ ____ _________ _______ _________.
________ _________ ___ __________ _______ _______ _______ ______ ______ __________.
______ ___ ___ ______ _________ ______ ______ ______ ____ _________ ___ ______.
_____ ____ ________ ______ _________ _________ ______ ______ ________ _____ ___.
____ __________ ________ _____ ________ ________ ___ ____ _________ _________.
_______ ____ ____ ___ ________ _______.
________ _______ _____ _________ __________ _________ _____ ____.
____ _______ ____ ____ ____ _______ ___ _____ _______.
________ ____ ___ ___ ______ __________ _______ _____ ________.
_____ _____ _______ ___ _____ ________.
_______ _____ _____ __________ ________ ____ _________ _______ __________ ________ ______ _______.
________ ______ ___ __________ ____ ___ ____ _____.
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a) If the partition P2 is a refinement of the partition P1 of [a,b], then and
. Verify this result for the function
defined over the interval
and the partitions
and
.
b) Evaluate: .
Prove that a strictly decreasing function is always one-one.
Determine the local minimum and local maximum values of the function f defined by .
Show that the function defined by
has an inverse by applying the inverse function theorem. Find its inverse also.
Show that is an algebraic number.
a) Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:
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Also check whether the function f is derivable at x = 1.
Verify Bozano-Weierstrass Theorem for the following sets:
i) Set of non-negative integers.
ii) Interval
d) Check whether the limit exists or not?