Verify the hypothesis and conclusions of the Fatou’s lemma for the sequence given by
We are asked to verify Fatou’s lemma for the sequence of functions [ f_n(x) = \begin{cases} 2n, & x \in \left(\frac{1}{2n}, \frac{1}{n}\right) [2mm] 0, & x \in \left(0, \frac{1}{2n}\right] \cup \left[\frac{1}{n}, 1\right] \end{cases} ] on the interval ([0,1]) with the Lebesgue measure. Let's go step by step. --- Step 1: Compute the integral of (f_n). [ \int_0^1 f_n(x),dx = ________ ___ ________ ________ ____ ______ ____ _______ ________ ____ __________.
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Let be given by
Show that f is locally invertible at all points in .
If E is a subset of with standard metric, then show that
.
Find the directional derivative of the function defined by
at the point (1, 2, -1, -2) in the direction .
Consider given by
Find f(2, 0, -1).
) 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.
1. State whether the following statements are true or false. Justify your answers.
a) The outer measure m^ of the set is 0.
b) A finite subset of a metric space is totally bounded.
c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.
d) The surface given by the equation can also be described by an equation of the form
in a neighbourhood of the point (0, 0).
e) A real valued function f on [a, b] is continuous if it is integrable on [a, b].
Does Cantor’s intersection theorem hold for the metric space with the standard metric? Justify your answer.