Fundamental Theorem of Analysis for discontinous functions

Last week I discovered a very neat property of improper Integrals. There are some functions where it is possible to calculate the area under the curve with the Fundamental Therorem of Analysis even if the function is discontinuous .

Let's say you have a function f(x)f\left( x \right) and you want to calculate abf(x)dx\int _{ a }^{ b }{ f\left( x \right) dx } . But unfortunately there is a pole point at xp{ x }_{ p } with b>xp>ab>{ x }_{ p }>a.Now I will prove that: If xp{ x }_{ p } is the only point on [a;b]\left[ a;b \right] where ff is discontinuos and there exists a point P(xp;yp)P\left( { x }_{ p };{ y }_{ p } \right) to which ff is point symmetric, then it is possible to calculate the integral with the Fundamental Therorem of Analysis.

Because it is an improper integral, you devide the ingral into two: abf(x)dx=limz0+(axpzf(x)dx+xp+zbf(x)dx)\int _{ a }^{ b }{ f\left( x \right) dx } =\lim _{ z\rightarrow { 0 }^{ + } }{ \left( \int _{ a }^{ { x }_{ p }-z }{ f\left( x \right) dx } +\int _{ { x }_{ p }+z }^{ b }{ f\left( x \right) dx } \right) } Because xp { x }_{ p } is the only point of discontinuity of ff, ff is continous on [a;xpz]\left[ a;{ x }_{ p }-z \right] and on [xp+z;b]\left[ { x }_{ p }+z;b \right] and therefore you can calculate this with the Fundamental Therorem of Analysis, if FF is the antiderivative of ff: abf(x)dx=limz0+(F(xpz)F(a)+F(b)F(xp+z))abf(x)dx=F(b)F(a)+limz0+(F(xpz)F(xp+z))\begin{aligned} \int _{ a }^{ b }{ f\left( x \right) dx } &= \lim _{ z\rightarrow { 0 }^{ + } }{ \left( F\left( { x }_{ p }-z \right) -F\left( a \right) +F\left( b \right) -F\left( { x }_{ p }+z \right) \right) } \\ \int _{ a }^{ b }{ f\left( x \right) dx } &= F\left( b \right) -F\left( a \right) +\lim _{ z\rightarrow { 0 }^{ + } }{ \left( F\left( { x }_{ p }-z \right) -F\left( { x }_{ p }+z \right) \right) } \end{aligned} If xp{ x }_{ p } is a pole point of ff then it is a pole point of FF as well. ff is point symmetric to P(xp;yp)P\left( { x }_{ p };{ y }_{ p } \right) and because ff is a measure of the slope of FF (because ddxF(x)=f(x)\frac { d }{ dx } F\left( x \right) =f\left( x \right) ), FF must be axially symmetrical to x=xpx={ x }_{ p }.

an example point symmetric function \(f\) in green and a possible axial symmetrical antiderivative of \(f\) in blue an example point symmetric function ff in green and a possible axial symmetrical antiderivative of ff in blue

And Therefore: F(xpz)=F(xp+z)F(xpz)F(xp+z)=0F\left( { x }_{ p }-z \right) =F\left( { x }_{ p }+z \right) \\ F\left( { x }_{ p }-z \right) -F\left( { x }_{ p }+z \right) =0

abf(x)dx=F(b)F(a)\int _{ a }^{ b }{ f\left( x \right) dx } =F\left( b \right) -F\left( a \right)

The geometric interpretation of this relationship is, that the infinite areas on the left and right side of the pole cancel out each other because of the opposite sign.

It is important to mention, that the antiderivative must be defined for every xxpx\neq { x }_{ p } in [a;b]\left[ a;b \right] , because otherwise it will not work.

Note by CodeCrafter 1
4 months, 1 week ago

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Edit:

In this proof, I used the argument, that the antiderivative of ff must be axial symmetrical to x=xpx={x}_{p}. But this is not true for all functions.

For example if f(x)=x3+1f \left( x \right) = x^3 + 1 then the antiderivative is F(x)=14x4+xF \left( x \right) = \frac{1}{4} x^4 + x. ff is point symmtric to P(0;1)P \left( 0;1 \right) but FF is not axial symmetrical to x=0x=0. But as we go closer and closer to the pole point, then we approach a "axial symmetry". Therefore near the pole point FF is axially symmetrical and hence the important limit in the proof above still converges to 0.

But unfortunately I cannot find a general argumentation, which guarantees that there are no counterexamples.

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