Which discontinuous functions are integrable?
There is a theorem that says that a function is integrable if and only if the set of discontinuous points has “measure zero”, meaning they can be covered with a collection of intervals of arbitrarily small total length.
Can an integral exist if it is discontinuous?
An improper integral of type 2 is an integral whose integrand has a discontinuity in the interval of integration [a,b]. This type of integral may look normal, but it cannot be evaluated using FTC II, which requires a continuous integrand on [a,b].
What function is integrable but not continuous?
Probably the simplest example of an integrable function that’s not continuous is something like f(x)={30≤x<151≤x≤2. This f is clearly not continuous at 1, but it is Riemann integrable on [0,2], with ∫20f(x) dx=8.
Can a discontinuous function be Riemann integrable if?
Discontinuous functions can be integrable, although not all are. Specifically, for Riemann integration (our normal basic notion of integrals) a function must be bounded and defined everywhere on the range of integration and the set of discontinuities on that range must have Lebesgue measure zero.
Are jump discontinuities integrable?
And a function with a (finite) jump discontinuity is integrable. And even some functions with infinite discontinuities are integrable.
Does a function have to be continuous for an integral?
Continuous functions are integrable, but continuity is not a necessary condition for integrability.
Are all integral functions continuous?
Continuity implies integrability; if some function f(x) is continuous on some interval [a,b], then the definite integral from a to b exists. While all continuous functions are integrable, not all integrable functions are continuous.
Are all functions integrable?
Many functions — such as those with discontinuities, sharp turns, and vertical slopes — are nondifferentiable. Discontinuous functions are also nondifferentiable. However, functions with sharp turns and vertical slopes are integrable.
Are all bounded function integrable?
Not every bounded function is integrable. For example the function f(x)=1 if x is rational and 0 otherwise is not integrable over any interval [a, b] (Check this). In general, determining whether a bounded function on [a, b] is integrable, using the definition, is difficult.
Is a discontinuous function differentiable?
If a function is discontinuous, automatically, it’s not differentiable.
Does integral exist if diverges?
Convergence and Divergence. If the limit exists and is a finite number, we say the improper integral converges . If the limit is ±∞ or does not exist, we say the improper integral diverges .