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nikolany
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Is there a function f:R-->R that is continuous at π and discontinuous at all other numbers?
Thx
Thx
Dick said:Consider the function f(x)=0 for x irrational and f(x)=x for x rational. Where is that continuous? Can you modify that for your purposes?
Yes, the classic example is the Dirichlet function, defined as f(x) = 1 for rational numbers and f(x) = 0 for irrational numbers. This function is continuous at only one point (x = 0) and discontinuous at all other numbers.
Yes, this is possible as shown by the Dirichlet function. A function can also be continuous at a finite number of points and discontinuous at all other numbers.
To prove that a function is discontinuous at all other numbers, we can use the definition of continuity. A function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. Therefore, to prove discontinuity at all other numbers, we need to show that the limit of the function at those numbers does not exist or is not equal to the function value.
Yes, these concepts are used in various fields such as economics, physics, and engineering. For example, in economics, the concept of a discontinuous function is used to model market crashes or unexpected changes in demand. In physics, the concept of a continuous function is used to model the smooth motion of objects, while discontinuous functions are used to model collisions or sudden changes in velocity.
No, a function cannot be both continuous and discontinuous at the same time. By definition, a function is either continuous or discontinuous at a point, but not both. However, a function can be continuous at one point and discontinuous at all other numbers.