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Swapnil
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What is the meaning of a pathological function?
HallsofIvy said:Often it refers to a function used as a counter-example for what appears to be an obvious statement. For example, there exist non-linear functions that satisfy f(x+y)= f(x)+ f(y) for all real x,y.
Actually, I can't give an example of such a function, but I can tell you how to construct one yourself!arildno said:Being non-linear, I would assume that this means there will exist non-integers a, so that for some x, we have the non-equality:
[tex]f(ax)\neq{af}(x)[/tex]
Could someone be generous enough to provide an example of such a function?
I'm curious..
matt grime said:It entirely depends on the context. Pathological in these contexts means, roughly, 'displaying some very bad behaviour that we can justifiably claim is not representative of the 'average' case'.
HallsofIvy said:Often it refers to a function used as a counter-example for what appears to be an obvious statement. For example, there exist non-linear functions that satisfy f(x+y)= f(x)+ f(y) for all real x,y.
matt grime said:And frequently to one thing that is a counter example to lots of related statements.
Chris Hillman said:In the last bullet, I quibble slightly with what matt grime said: in my experience, particularly in the context of discoveries by nineteenth century analysts, "pathological functions" are always "counterintuitive" but their bad behavior often turns out to be, in some sense, "generic".
Cool!HallsofIvy said:Actually, I can't give an example of such a function, but I can tell you how to construct one yourself!
Consider the real numbers as a vector space over the rational numbers. There must exist a basis for such a vector space (there exist a basis for any vector space over any field) but the basis clearly must be uncountable so I can't give you an example. I can, however, assert that 1 and e, for example, are independent and so I can construct a basis containing 1, e, and uncountably many other numbers. Define f(1)= 1, f(e)= 2, f(x)= 0 for x any "basis" number other than 1 or e. Define f(x) for any other x "by linearity". That is, write x as a linear combination a1(1)+ a2(e)+ ... Then f(x)= a1+ 2a2. That, since it is a linear function over the vector space, satisfies f(x+ y)= f(x)+ f(y). It also satisfies f(qx)= qf(x) where q is any rational number but not for q equal to an irrational number. In particular, it is not of the form f(x)= Cx and, so, can be shown not to be continuous.
That is a really "pathological" function! If you attempted to draw its graph, you would have to blacken the entire paper. I don't mean by that that its graph contains every point in the plane. It is, after all, a function and so if crosses any vertical line only once. However, what ever pencil or pen you use to draw the graph has a point with some non-zero radius. What is true is that the graph is "dense in the plane"- every point in the plane is within distance [itex]\delta[/itex] of a point of the graph for any [itex]\delta> 0[/itex].
Pathological function refers to any abnormal or dysfunctional process that occurs in the body, leading to disease or illness. It is often caused by genetic, environmental, or lifestyle factors.
While normal function refers to the healthy and efficient functioning of the body's systems, pathological function involves disruptions or malfunctions in these systems, resulting in disease or dysfunction.
Some examples of pathological function include autoimmune diseases, such as rheumatoid arthritis, infections caused by bacteria or viruses, and genetic disorders, such as cystic fibrosis. Mental health disorders, such as depression and anxiety, can also be considered pathological functions.
Pathological function is diagnosed through various methods, including medical history, physical exams, laboratory tests, and imaging scans. Treatment may involve medication, surgery, therapy, or lifestyle changes, depending on the specific condition and its severity.
Some forms of pathological function, such as genetic disorders, cannot be prevented. However, many diseases and illnesses caused by pathological function can be prevented through healthy lifestyle choices, such as maintaining a balanced diet, exercising regularly, and avoiding harmful substances.