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What was Georg Bernard Riemann's example of a function that is continuous at zero and irrational values but discontinuous at positive and negative rational numbers?
Wiki User
∙ 16y ago
f(x)=0 if x=0 or x is irrational, f(x)=1/q if x=p/q "in lowest terms", i.e., gcd(p,q)=1.
To see that it satisfies the claims, you simply need to verify that the limit as x approaches p is 0 for all p, which is pretty easy. From this both claims follow.
Wiki User
∙ 16y ago
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Can a discontinuous function have a laplace transform?
Sure! The definition of Laplace transform involves the integral of a function, which always makes discontinuous continuous.
Is the infinite sum of continuous function continuous?
An infinite sum of continuous functions does not have to be continuous. For example, you should be able to construct a Fourier series that converges to a discontinuous function.
Can a discontinuous function be approximated by a fourier series?
Yes it can.
What does it mean for a function to be continuous?
It means that the function value doesn't make sudden jumps. For example, a function that rounds a number down to the closest integer is discontinuous for all integer values; for instance, when x changes from 0.99 to 1, or from 0.999999 to 1, or for any number arbitrarily close to 1 (but less than one) to one, the function value suddenly changes from 0 to 1. At other points, the function is continuous.
Does the cosine function appear to be continuous?
yes it is a continuous function.
Can a discontinuous function have a laplace transform?
Sure! The definition of Laplace transform involves the integral of a function, which always makes discontinuous continuous.
Is the infinite sum of continuous function continuous?
An infinite sum of continuous functions does not have to be continuous. For example, you should be able to construct a Fourier series that converges to a discontinuous function.
Where is f(x) discontinuous but not differentiable Explain?
Wherever a function is differentiable, it must also be continuous. The opposite is not true, however. For example, the absolute value function, f(x) =|x|, is not differentiable at x=0 even though it is continuous everywhere.
Discontinuous function in fourier series?
yes a discontinuous function can be developed in a fourier series
What is X2-4 divided by x2 plus 3x plus 2 continuous or discontinuous?
The numerator function x2 - 4 and the denominator function x2 + 3x + 2 are both continuous functions of x for the entire x-axis. However, the quotient of these two functions is not continuous when the denominator function has the value of 0, because division by zero is not defined. The denominator function is 0 when x = -1 or -2. Therefore, the quotient function is not fully continuous over any intervals that include -1 or -2, but it is "piecewise continuous" over other intervals of the x-axis.
How do you sketch a piece wise continuous function?
A piece-wise continuous function is one which has a domain that is broken up inot sub-domains. Over each sub-domain the function is continuous but at the end of the domain one of the following possibilities can occur:the domain itself is discontinuous (disjoint domains),the value of the function is not defined at the start or end-point of the domain ((a hole),the value of the function at the end point of a sub-domain is different to its value at the start of the next sub-domain (a step-discontinuity).A piece-wise continuous function is one which has a domain that is broken up inot sub-domains. Over each sub-domain the function is continuous but at the end of the domain one of the following possibilities can occur:the domain itself is discontinuous (disjoint domains),the value of the function is not defined at the start or end-point of the domain ((a hole),the value of the function at the end point of a sub-domain is different to its value at the start of the next sub-domain (a step-discontinuity).A piece-wise continuous function is one which has a domain that is broken up inot sub-domains. Over each sub-domain the function is continuous but at the end of the domain one of the following possibilities can occur:the domain itself is discontinuous (disjoint domains),the value of the function is not defined at the start or end-point of the domain ((a hole),the value of the function at the end point of a sub-domain is different to its value at the start of the next sub-domain (a step-discontinuity).A piece-wise continuous function is one which has a domain that is broken up inot sub-domains. Over each sub-domain the function is continuous but at the end of the domain one of the following possibilities can occur:the domain itself is discontinuous (disjoint domains),the value of the function is not defined at the start or end-point of the domain ((a hole),the value of the function at the end point of a sub-domain is different to its value at the start of the next sub-domain (a step-discontinuity).
Can a discontinuous function be approximated by a fourier series?
Yes it can.
What does it mean for a function to be continuous?
It means that the function value doesn't make sudden jumps. For example, a function that rounds a number down to the closest integer is discontinuous for all integer values; for instance, when x changes from 0.99 to 1, or from 0.999999 to 1, or for any number arbitrarily close to 1 (but less than one) to one, the function value suddenly changes from 0 to 1. At other points, the function is continuous.
Is a cosine function continuous?
Yes. The cosine function is continuous. The sine function is also continuous. The tangent function, however, is not continuous.
A function that is discontinous everywhere?
An example of a function discontinuous everywhere fa(x) = 1 , if x belongs to set a fa(x) = 0 if x does not belong to set a The a is a subscript here. This function is called the 'Indicator function' and everywhere discontinuous.
Is the identity function bounded or unbounded?
That depends! The identity operator must map something from a space X to a space Y. This mapping might be continuous - which is the case if the identify operator is bounded - or discontinuous - if the identity operator is unbounded.
Does the cosine function appear to be continuous?
yes it is a continuous function.
