Table of Contents
Does a function have to be differentiable to be integrated?
Even if you’re integrating over a bounded domain, differentiability is neither necessary nor sufficient. Take on . If you’re integrating over a segment however (closed and bounded), or more generally, a compact set, then differentiability implies continuity which in turn implies integrability.
Does integrable imply differentiable?
In practical terms, integrability hinges on continuity: If a function is continuous on a given interval, it’s integrable on that interval. This is just one of infinitely many examples of a function that’s integrable but not differentiable in the entire set of real numbers.
Can a function be differentiable but not continuously differentiable?
We see that if a function is differentiable at a point, then it must be continuous at that point. There are connections between continuity and differentiability. If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on .
Is a constant function infinitely differentiable?
For example, a constant function is infinitely differentiable – all of its derivatives are zero. A quadratic polynomial is infinitely differentiable – its third and higher order derivatives are all zero.
Can a function be integrated if it is not continuous?
And integral of every continuous function continuous? According to me answer of second part is yes as integration simply means area under curve.
Can we integrate every function?
Not every function can be integrated. Some simple functions have anti-derivatives that cannot be expressed using the functions that we usually work with.
Does integrable imply continuous?
An integrable function is not necessarily continuous. Discontinuous functions can be both integrable and non integrable and continuity isn’t guaranteed even if a function is integrable, on some [a,b].
Is differentiable function always continuous?
A differentiable function is necessarily continuous (at every point where it is differentiable). It is continuously differentiable if its derivative is also a continuous function.