Can a continuous function go to infinity?
Yes, you can make your function go from R to the “extended real numbers” {−∞}∪R∪{∞}, a topological space that is homeomorphic to [0,1], using a topology that should be pretty obvious. Then if you define f(0)=∞, your function is continuous at 0.
Why wave function vanishes at infinity?
For the probability interpretation of wave functions to work, the latter have to be square integrable and therefore, they vanish at infinity. The idea is they approximate what is physically achievable but are mathematically more tractable. For example in QM a particle with exact momentum is a wave-function to infinity.
What does it mean if a function vanishes?
zero
A quantity which takes on the value zero is said to vanish. For example, the function vanishes at the point. . For emphasis, the term “vanish identically” is sometimes used instead, meaning the quantity in question does not merely vanish by all appearances, but is mathematically identically equal to zero.
What does it mean for a function to never vanish?
In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. This definition can be formalized in many cases by adding an (actual) point at infinity.
What types of functions are always continuous on − ∞ ∞?
Every polynomial function is continuous everywhere on (−∞, ∞). (ii.) Every rational function is continuous everywhere it is defined, i.e., at every point in its domain.
Are wave functions infinite?
The mathematical representations of the wavefunctions extends to infinity since there are no boundary conditions to limit the distance.
Why is wave function zero when potential is infinite?
The potential is taken to be infinite for educational reasons. This forces the wave function to be zero at the boundary and outside the box, for finite energy solutions, and simplifies the problem of solving the wave equation considerably.
What is a vanishing denominator?
A map with vanishing denominator is a map of ℝn with n ≥ 2, that is not defined in the whole space ℝn because at least one of its components contains a denominator which can vanish. To make simpler the notation we only refer to the case where only one component of the map is characterized by a vanishing denominator.
What does vanishing mean in linear algebra?
of a number, quantity, or function) to become zero.”
What is compact support of a function?
A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact support if its support is a compact set. For example, the function in its entire domain (i.e., ) does not have compact support, while any bump function does have compact support.
Which types of functions are always continuous on − ∞ ∞?