IGNOU MTE 9 2026 - Are the following statements true or false? Give reasons for your answ

Question

Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]0, 1] is an open set.
b) Every bounded sequences is not convergent.
c) The function $f : [-2, 2] \to \mathbf{R}$ defined by $f(x) = \frac{4x + 3}{x^2 + 1}$ is uniformly continuous.
d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.
e) The function $f : [0, 2] \to \mathbf{R}$ defined by $f(x) = x + [x]$ is not integrable.

24 Mar 2026

Answer :

Word Count : 198

a) False. The complement of the interval ]0, 1] in ℝ is (−∞, 0] ∪ (1, ∞), which contains closed endpoints 0 and 1. Since it includes its boundary points, it is not an open set. b) True. A bounded sequence need not be convergent. For example, the sequence (a_n = (-1)^n) is bounded between −1 and 1 but does not approach any single limit, hence it is not

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