Show that if f is a continuous real-valued function on the compact space X,
then there exist points x_1, x_2 in X such that
f(x_1)=inf{f(x):x in X} and
f(x_2)=sup{f(x):x in X}.

Solution Summary

The solution contains the proof of the theorem "if f is a continuous real-valued function on the compact space X,
then there exist points x_1, x_2 in X such that
f(x_1)=inf{f(x):x in X} and
f(x_2)=sup{f(x):x in X}".

... and beta: X_2 ---> X_3 are continuous functions of compact... Z is a homeomorphism of compact topological spaces... 3. We consider a continuous function , where and ...

... It is proved that for a compact Hausdorff space X, and a continuous function f on X, there is a fixed set for f, that is, a nonempty subset A of X such that ...

... and is compact with Consider the identity function defined by ... Now is continuous if, for every closed subset is ... every closed subset of a compact space is compact...

... Let X be a set and F is separating collection of functions f: X ... the weakest or smallest topology on X for which each function in F is continuous (see attached ...

... the same as the definition of uniform convergence of a sequence of functions (see http ... Theorem: If set is compact and real function is continuous on , then ...

... for the cells to carry out its bodily functions. ... Define the function, general structure, and give examples ... contain cells arranged in continuous sheets ( either ...

... and by adding a spatial objec- functions and spatial ... Creating continuous areas of old ing models. ... guesses to variables and objective function exactly, the ...

... From this, we conclude n=1 n=1 that Y is compact. ... subsets in the relative (or intrinsic) topology of a space is a ... Let f : Rn → Rn be a continuous function. ...

... 3. Intrapsychological functioning. ... according to Vygotsky does not follow a continuous and linear ... a clearly diﬀerent, new and independent function of speech ...

... Multiple trial functions are used to enrich the ... subproblem for any given forcing function by projecting ... categorize data as either categorical or continuous. ...