Integrability of Monotonic Functions on Closed Intervals Explained

[Miike012]

The book is saying if f is monotonic on a closed interval, then f is integrable on the closed interval.

Or basically if it is increasing or decreasing on the interval it is integrable on that interval

This makes sense, however this theorem seems to obvious because obviously if a function is countinuous on a closed interval it will be integrable on that interval whether or not its increasing or not...
So my question is... what is a non monotonic function..? would that be a function with discountinuities?

 

[LCKurtz]

The book is saying if f is monotonic on a closed interval, then f is integrable on the closed interval.

Or basically if it is increasing or decreasing on the interval it is integrable on that interval

This makes sense, however this theorem seems to obvious because obviously if a function is countinuous on a closed interval it will be integrable on that interval whether or not its increasing or not...
So my question is... what is a non monotonic function..? would that be a function with discountinuities?

It doesn't have to be discontinuous. For example f(x) = xsin(1/x) is continuous if you define f(0) = 0, but it isn't monotonic on any closed interval containing 0. A discontinuous example is the "salt and pepper" function g(x) = 1 if x rational and 0 if x irrational, which is not monotonic on any interval.

 

[Miike012]

Sorry, I don't see how f(x) = xsin(1/x) is defined at x = 0...?

 

[LCKurtz]

Sorry, I don't see how f(x) = xsin(1/x) is defined at x = 0...?

The definition of the function I am suggesting is$$
f(x) = \begin{cases} \frac 1 x\sin(x)&x \neq 0\\
0 & x = 0\end{cases}$$ It is defined to be 0 when x = 0.

 

[Dick]

The book is saying if f is monotonic on a closed interval, then f is integrable on the closed interval.

Or basically if it is increasing or decreasing on the interval it is integrable on that interval

This makes sense, however this theorem seems to obvious because obviously if a function is countinuous on a closed interval it will be integrable on that interval whether or not its increasing or not...
So my question is... what is a non monotonic function..? would that be a function with discountinuities?

f(x)=x^2 on [-1,1] is nonmonotonic. It is continuous. It's also integrable. What's the question again? I think Miike012 might be confusing f monotonic -> f integrable (which is true) with f not monotonic -> f not integrable (which is false).