Teaching CSIR NET Mock Test Series Mathematical Science Analysis Discontinuity & Functions of Bounded Variation
Which of the following statements are true?
1
The function f : ℝ → ℝ defined by \(f(x)=\left\{\begin{array}{cl} {[x] \sin \frac{1}{x}} & \text { for } x \neq 0, \\ 0 & \text { for } x=0 \end{array}\right.\) has a discontinuity at 0 which is removable.
2
The function f : [0, ∞) → ℝ defined by \(f(x)=\left\{\begin{array}{cl} \sin (\log x) & \text { for } x \neq 0, \\ 0 & \text { for } x=0 \end{array}\right.\)has a discontinuity at 0 which is NOT removable.
3
The function f : ℝ → ℝ defined by \(f(x)= \begin{cases}e^{1 / x} & \text { for } x<0, \\ e^{1 /(x+1)} & \text { for } x \geq 0\end{cases}\)has a jump discontinuity at 0.
4
Let f, g : [0, 1] → ℝ be two functions of bounded variation. Then the product fg has at most countably many discontinuities.