I posted this problem before this .I have satisfied explaination given by markus-scheuer sir and siminore sir . I also found here .
I have read the Wikipedia posts for continuous function and bounded function and I have concluded that a function should be defined on given range for satisfying conditions of continuity and bounded function .
I posted my answer at GateOverflow(a site old Q/A of GATE ) , but they not accepted my answer and they have given reason answer key is correct , but not my explanation .
Let $f(x)=x^{-(1/3)}$ and $A$ denote the area of region bounded by $f(x)$ and the $X-$axis, when $x$ varies from $-1$ to $1$. Which of the following statements is/are TRUE?
- $f$ is continuous in $[-1, 1]$
- $f$ is not bounded in $[-1, 1]$
- $A$ is nonzero and finite
Given answer key of this question $:$
- False
- True
- True
In my opinion all given statement should be false .
My doubts are $:$
- If a function is not continuous then is it possible for bounded function in given range ? What about function in given problem .
- Is it possible bounded area zero or infinite ? What about statement $(3)$ of problem ?