67859 Maximum and Minimum theorem Maximum and Minimum theorem Please see the attachment. Prove: A continuous mapping T of a compact subset M of a metric space X into assumes a maximum and a minimum at some points of M.
28575 Extreme Value Theorem Extreme Value Theorem To prove that f attains maximum and minimum, we need the following lemmas. Lemma 1: f:K->R continuous, K compact in R, F(K) is compact.
Then f has an absolute maximum and an absolute minimum on [a, b] 1 (d) Does the conclusion of the Max-Min Theorem always hold for a *bounded* function f:R ->R that is continuous on R? Prove or give a counterexample.
To prove it, we can make use of two theorems from topology. Theorem 1. Heine-Borel Theorem. A set is compact if and only if E is closed and bounded. Theorem 2. Let be continuous with E compact. Then is compact.
Thus is also an identity function on . 7. Proof: 8. Proof: If is a homeomorphism of compact topological spaces, according to the results of 4 and 5, is a linear and continuous. Along with the result of 6, is also a homeomorphism.
Since X is a compact metric and f_n blongs to C(X,Y), then f_n is continuous and thus f_n-f_o(x) is continuous. We know, on a compact metric, a continuous function can reach its maximum and minimum value.
Prove that the when the interval I is compact, the set X defined above of normed spaces coincides with C(I,X), and that the sup norm and the norm ||.||_epsilon,h are equivalent.
Theorem III: Let S be the bounded and close point set, and let S be the function defined on S which is continuous at each point of S. then the values of f are bounded. Proof: Let S be the bounded and close point set.
But we can find a y' in [0,1] such that: if y is in a small neighbourhood of y'. In fact, f is continuous on a compact set, so the maximum is always attained at some point of the unit square.
It is shown that if f is an upper semi-continuous function on a compact subset K of R^p with values in R, then f is bounded above and attains its supremum on K. The solution is detailed and well presented.