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Nusc
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What is an example where it's Riemann integrable int(f(t),t,a,x) but no derivative exists at certain pts?
HallsofIvy said:Those two examples also have the property that while [itex]F(x)= \int f(t)dt[/itex] is defined, F(x) itself has no dervative at x= 0.
Yes, the function f(x) = |x| is Riemann integrable on the interval [-1,1], but it has no derivative at x=0.
This is because the Riemann integral only requires that a function be continuous on a closed interval, but a derivative requires the function to be continuous on an open interval.
Yes, the function f(x) = |x|sin(1/x) is Riemann integrable on the interval [0,1], but it has no derivative at x=0.
Yes, this is possible. For example, the function f(x) = x^2sin(1/x) has a derivative at all points except x=0, but it is still Riemann integrable on the interval [0,1].
To determine if a function is Riemann integrable but has no derivative at certain points, you can use the Riemann criterion, which states that a function is Riemann integrable if and only if it is continuous almost everywhere. This means that a function can have a few isolated points where it is not continuous and still be Riemann integrable.