If f is integrable on [a, b], show by example that

Question

show by example that:

t

F (t) = ∫ f(x) dx

a

is continuous but need not to be differentiable.

note: the integral from (a) to (t).

Answer 1

So there are actually absolute value functions that are continuous but not differentiable everywhere. Take for example f(x) = |x|. The point (0,0) isn't differentiable in that function because it's not a smooth transition; it's a sharp turn. So you can just replace the f(x) with |x|. In order to actually calculate the integral, you would need to see if the undifferentiated point is included in the interval. If it isn't, then just take away the absolute value and integrate normally. If it is, then you would need to create multiple integrals based on how many undifferentiated points there are and then add them together. In this case, there's only one, so you would just need to create two integrals, one bounded from the lowest bound to the undifferentiated point and one bounded from the undifferentiated point to the highest bound. Then you would need to test if a number within each interval evaluates to a positive or negative number. If it's positive, then leave the integral alone. If it's negative, then put the negative sign in front of the interval. Then you evaluate from there. That's how you would integrate an absolute value function.