Integrability criteria help us figure out which functions we can actually integrate. They're like a checklist for determining if a function is well-behaved enough to work with. This is crucial for understanding the Riemann integral.
These criteria cover different types of functions, from smooth and continuous ones to those with jumps or wild behavior. Knowing these rules helps us tackle a wide range of integration problems and understand the limits of Riemann integration.
Integrability of Monotonic Functions
Definition and Properties of Monotonic Functions
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A function is monotonic on an interval if it is either non-increasing or non-decreasing on that interval
Examples of monotonic functions include linear functions (f(x)=mx+b) and power functions (f(x)=xn for n∈Z)
Monotonic functions have the property that for any two points x1,x2 in the interval, if x1<x2, then either f(x1)≤f(x2) (non-decreasing) or f(x1)≥f(x2) (non-increasing)
The sum, difference, product, and quotient of two monotonic functions with the same monotonicity are also monotonic
Integrability and Integration of Monotonic Functions
A monotonic function on a closed interval is always integrable on that interval
The integrability of a monotonic function on a closed interval can be proven using the properties of Riemann sums and the fact that a monotonic function has a finite number of discontinuities
The integral of a monotonic function on a closed interval can be calculated using the Fundamental Theorem of Calculus, even if the function is not continuous
For example, the function f(x)=⌊x⌋ (greatest integer function) is monotonic and discontinuous on [0,1], but its integral can be calculated as ∫01⌊x⌋dx=0
Integrability of Continuous Functions
Definition and Properties of Continuous Functions
A function is continuous on a closed interval if it is continuous at every point in the interval, including the endpoints
Examples of continuous functions include polynomials, exponential functions (f(x)=ex), and trigonometric functions (sine, cosine)
Continuous functions have the intermediate value property if f(a)<y<f(b), then there exists a point c∈(a,b) such that f(c)=y
The sum, difference, product, and quotient (when defined) of two continuous functions are also continuous
Integrability and Integration of Continuous Functions
The Extreme Value Theorem states that a continuous function on a closed interval attains its maximum and minimum values on that interval
The integrability of a continuous function on a closed interval can be proven using the properties of Riemann sums and the Extreme Value Theorem
The integral of a continuous function on a closed interval can be calculated using the Fundamental Theorem of Calculus
For example, the function f(x)=sin(x) is continuous on [0,π], so its integral can be calculated as ∫0πsin(x)dx=2
Integrability of Discontinuous Functions
Integrability of Functions with Finite Discontinuities
A function with a finite number of discontinuities on a closed interval is integrable on that interval
The integrability of a function with a finite number of discontinuities can be proven by partitioning the interval into subintervals where the function is continuous and applying the integrability of continuous functions on closed intervals
The integral of a function with a finite number of discontinuities can be calculated by splitting the integral into a sum of integrals over the subintervals where the function is continuous
For example, the function f(x)={x20x=1x=1 has a removable discontinuity at x=1, but is integrable on [0,2]
Types of Discontinuities
Removable, jump, and infinite discontinuities are the three types of discontinuities that a function may have
A removable discontinuity occurs when the limit of the function exists at the point, but the function is not defined or has a different value at that point (f(x)=x−1x2−1 at x=1)
A jump discontinuity occurs when the left-hand and right-hand limits of the function at the point exist but are different (f(x)=⌊x⌋ at integer values)
An infinite discontinuity occurs when the limit of the function at the point is infinite (f(x)=x1 at x=0)
Functions with these types of discontinuities can still be integrable on a closed interval
Lebesgue Criterion for Integrability
Statement and Implications of the Lebesgue Criterion
The Lebesgue criterion states that a bounded function on a closed interval is Riemann integrable if and only if the set of its points of discontinuity has measure zero
A set has measure zero if, for every ε>0, the set can be covered by a countable collection of intervals whose total length is less than ε
The Lebesgue criterion provides a necessary and sufficient condition for the integrability of bounded functions on closed intervals
This criterion implies that functions with "small" sets of discontinuities (in terms of measure) are integrable, while functions with "large" sets of discontinuities are not
Examples of Functions Satisfying or Failing the Lebesgue Criterion
Functions that satisfy the Lebesgue criterion include continuous functions, monotonic functions, and functions with a finite number of discontinuities
For example, the function f(x)={10x∈Qx∈/Q (Dirichlet function) has a set of discontinuities equal to R, which has positive measure, so it is not Riemann integrable on any interval
The function f(x)={x0x∈Qx∈/Q has a set of discontinuities equal to R∖Q, which has measure zero, so it is Riemann integrable on any closed interval
The Lebesgue criterion demonstrates the connection between the integrability of a function and the "size" of its set of discontinuities, providing a powerful tool for analyzing the integrability of bounded functions