C^infty on U refers to a class of functions that are infinitely differentiable on a given open set U in a mathematical space. This means that the functions have derivatives of all orders on U.
A function is infinitely differentiable if it has derivatives of all orders at every point in its domain. This means that the function is smooth and has a well-defined slope at every point.
C^infty on U is an important concept in mathematical analysis and differential geometry. It allows for the study of smooth and continuous functions that have derivatives of all orders, which are essential in many fields such as physics, engineering, and economics.
Yes, the function f(x) = |x| is not infinitely differentiable on the open set U = (-1,1). This is because the function is not differentiable at x = 0, as it has a sharp corner at that point.
C^infty on U functions can be represented by their Taylor series, which is an infinite sum of terms that approximate the function at a given point. This series is useful in approximating the value of a function and its derivatives at any point within its domain.