Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.
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Let be given by
Show that f is locally invertible at all points in .
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].
Let (X, d) be a metric space and A be a subset of X. Show that if and only if A is both open and closed.
For the equation , at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form
.
Find the interior, closure, the set of limit points and the boundary of the set
in with the standard metric.
Find the directional derivative of the function defined by
at the point (1, 2, -1, -2) in the direction .
If E is a measurable set and f is a simple function such that , show that