Show that the function f' if it exists in an interval, can not have an ordinary or removable discontinuity in that interval.

The function f(x) = x^x decrease on the interval

The function f(x) = x^2 is increasing in the interval

The function f(x)= tan^(-1)x +x increases in the interval

The function f(x) is a decreasing funciton in the interval

The function f(x)=x^(1//x) is increasing in the interval

If both Lim_(xrarrc^(-))f(x) and Lim_(xrarrc^(+))f(x) exist finitely and are equal, then the function f is said to have removable discontinuity at x=c . If both the limits i.e. Lim_(xrarrc^(-))f(x) and Lim_(xrarrc^(+))f(x) exist finitely and are not equal, then the function f is said to have non-removable discontinuity at x=c . Which of the following function not defined at x=0 has removable discontinuity at the origin?

The function f(x)=x/(xlogx) increase on the interval