function $f(x)=x^{-(1/3)}$ , please check whether my solution is correct? 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?


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*f is continuous in [-1, 1]

*f is not bounded in [-1, 1]

*A is nonzero and finite



I try to explain :

graph is  :



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*False , since left limit is not equivalent to right limit.

*True , since  f(x) rises to $infinite$ at $x=0$.

*True , since we can calculate it by doing integrating the function


The following statements certainly have standard definitions: 1) "$f$ is continuous at $x$" ; 2) "$f$ is continuous".  For the function $f(x) = x^{-1/3}$, it's certainly true that "$f$ is continuous" and that "$f$ is not continuous at $0$".
However, what about the statement, "$f$ is continuous on $[a,b]$"?  Perhaps there is some confusion or disagreement over the meaning of this statement.  I think that the statement "$f$ is continuous on $[a,b]$" means that "if $x \in [a,b]$ then $f$ is continuous at $x$".
By that definition, statement 1) is false because the function $f(x) = x^{-1/3}$ is not continuous at $0$.
So, for $f(x) = x^{-1/3}$, the statement "$f$ is continuous" is true, but the statement "$f$ is continuous on $[-1,1]$" is false.
