Find the Lebesgue integral of the function f given by
The function ( f ) is given as [ f = \chi_{[0,1]} + 2 \chi_{(3,5]} + \chi_{[6,9]}. ] The Lebesgue integral of a simple function that is a _________ ________ _________ ________ ______ _________ __________ ____.
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) Show that the function f defined by
is not differentiable at (0,0). Do the partial derivatives of f exist at (0,0)? or at any other point in ? Justify your answer.
Find the Lebesgue integral of the function f given by
Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.
Give an example of a family fi of subsets of a set X which has finite intersection property. Justify your choice of example.
Which of the following subsets of are compact w.r.t. the metric given against them. Justify your answer.
i) in
of
with standard metric.
ii) with discrete metric.
iii) with standard metric.
Find the interior and closure of the set of rationals in
with standard metric.
Find the interior, closure, the set of limit points and the boundary of the set
in with the standard metric.