If f and g be continuous real valued functions on the metric space M. Let A be the set of all x ∈ M s.t. f(x)

Question

If f and g be continuous real valued functions on the metric space M. Let A be the set of all x &isin; M s.t. f(x) < g(x)a)A is closedb)A is openc)Neither open nor closedd)None of theseCorrect answer is option 'B'. Can you explain this answer?

Answer

Understanding the Set A
The set A is defined as all points x in the metric space M where f(x) < g(x).='' to='' analyze='' the='' properties='' of='' this='' set,='' we='' consider='' the='' functions='' f='' and='' />
Continuity of Functions
- Since f and g are continuous functions, the difference h(x) = g(x) - f(x) is also continuous.
- The continuity of h means that small changes in x lead to small changes in h(x).
Characterizing Set A
- The set A can be rewritten as A = { x in M | h(x) > 0 }.
- This is the preimage of the open set (0, ∞) under the continuous function h.
Properties of Open Sets
- In topology, the preimage of an open set under a continuous function is open.
- Since (0, ∞) is open in the real numbers, the preimage A must also be open in the metric space M.
Conclusion
- Therefore, the set A, where f(x) < g(x),='' is='' indeed='' an='' open='' />
- Hence, the correct answer is option 'B', indicating that A is open.
By understanding the properties of continuous functions and their implications on the sets they define, we can conclude that the set A is open in the given metric space.

Similar Mathematics Doubts

If f(x) = 3x - 2 and (gof)-1 (x) = x - 2, then function g(x) is

Let S be a closed set of R, T a compact set of R such that S T. ThenS T is

Let S = {x R : x6 x5 100} and T = {x2 2x : x (0, )}. Then set S T is

Letwhere [] denotes the greatest integer function. If f(x) is continuous at x = 2 then the sum of the values of a and b is______. Correct answer is '-2'. Can you explain this answer?

Let S be the surface of cone z=√x^2 y^2 bounded by the planes z=0 and z=3. further, let C be the closed curve forming the boundary of the surface S. A vector field F is such that curl F =-xi-yj .the absolute value of the line integral F.dr?

Top Courses for Mathematics

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams