Question
If f(x) is continuous for all real values of
x and f(x) takes on only rational values, then if f(1)=1, the value of f(0) isSolution
Property: - Let f:[0,1]→ R be continuous such that f(x)∈Q for any x∈[0,1] then f (x) is constant. Proof Suppose f isn't constant. Then for some a,b∈[0,1], f(a)≠f(b); Without Loss of Generality, f(a) Since f is continuous, by the Intermediate Value Theorem, it must take every value in the interval [f(a), f(b)]. But this interval contains an irrational number (in fact, uncountably many of them). Contradiction. Hence, f is constant and equal to 1. Therefore, Since f(x) can take only rational values, option c
When we talk about Demographic Transition, which Stage is characterised with high birth rate, high number of deaths?
Which of the following gives the recommendations to the government on top-level appointments like full-time Directors, non-Executive Chairman in Public...
XYZ Ltd. is planning a private placement to raise capital and is considering including the following groups:
a. 40 identified individuals.
Which of the following is true regarding the Reserve Bank of India's prudential norms for income recognition, asset classification, and provisioning fo...
Calculate Proprietary Ratio
Which of the following is not one of the major economic challenges that posed threat to global growth, according to the Economic Survey of 2023?
What are the limitations on the appointment and reappointment of auditors for listed companies under the Companies Act?
Which city did American Express open its largest campus worldwide?
BCR is the ratio for determining viability of a project. The project is viable when BCR is one or more than one and is unviable when it is less than one...
Which category do Bad debt fall under?