Tag : function of one variable
- For x ∈ R, let ⌊x⌋ denote the greatest integer less than or equal to x. For x, y ∈ R, define min{x,y} = {(x, if x le y),(y, otherwise.):} Let f: [-2pi, 2pi] → R be defined by f(x) = sin(min{x, x - ⌊x⌋}) for x ∈ [-2pi, 2pi]. Consider the set S = {x ∈ [-2pi, 2pi]: f is discontinuous at x}. Which one of the following statements is TRUE?
- Consider the following two statements. P: There exist functions 𝑓: ℝ → ℝ, 𝑔: ℝ → ℝ such that 𝑓 is continuous at x = 1 and 𝑔 is discontinuous at 𝑥 = 1 but 𝑔 ∘ 𝑓 is continuous at 𝑥 = 1. Q: There exist functions 𝑓: ℝ → ℝ, 𝑔: ℝ → ℝ such that both 𝑓 and 𝑔 are discontinuous at 𝑥 = 1 but 𝑔 ∘ 𝑓 is continuous at 𝑥 = 1. Which one of the following holds?
- Let 𝑓: ℝ → ℝ be defined by f(x) = (x^2 + 1)^2/(x^4 + x^2 + 1) for x ∈ ℝ. Then, which one of the following is TRUE?
- Let S = {f : R → R : f is polynomial and f(f(x)) = (f(x))^2024 for x ∈ R}. Then, the number of elements in S is _____________