Are all differentiable functions infinitely differentiable?

Not all functions are infinitely differentiable. Trivially we see that not all functions are differentiable even one time. For example, the Weierstrass function is everywhere continuous, yet nowhere differentiable. Clearly a function is not infinitely differentiable if it is not even once differentiable.

How do you show that a function is infinitely differentiable?

f is continuously differentiable or C1 if f is differentiable and the function f is continuous. f is infinitely differentiable or C∞ if the nth derivative f(n) is de- fined on all of U for all positive integers n. with the right hand side absolutely convergent. f(x) := { x2 sin(1/x), x = 0, 0, x = 0.

What function is not infinitely differentiable?

Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. There are however stranger things. The function sin(1/x), for example is singular at x = 0 even though it always lies between -1 and 1.

Are all infinitely differentiable functions analytic?

As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or C∞). (Note that this differentiability is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions that are not analytic: see non-analytic smooth function.

What is the meaning of infinitely differentiable?

in most situations, infinitely differentiable means that you are allowed to differentiate the function as many times as you wish, since these derivatives exist (everywhere). The value of the derivatives is irrelevant, of course.

Is tangent infinitely differentiable?

Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency.

How do you show a function is differentiable?

A function is said to be differentiable if the derivative of the function exists at all points in its domain. Particularly, if a function f(x) is differentiable at x = a, then f′(a) exists in the domain. Let us look at some examples of polynomial and transcendental functions that are differentiable: f(x) = x4 – 3x + 5.

Does smooth mean infinitely differentiable?

In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth.

Are constant functions analytic?

Constant functions are analytic.

Is a constant function smooth?

A smooth function has to be continuous and differentiable to any order at every point in its domain. In other words it has to be continuously differentiable to any order.

What is non smooth?

Non-smooth in the context of optimization usually means that the cost or constraints are not differentiable. The l1,l∞ norms are examples. The l1 norm is not differentiable on the axes, the l∞ is not differentiable on the ‘diagonals’. Another, less trivial, is the maximum singular value of a matrix.

What is not differentiable?

A function is not differentiable at a if its graph has a vertical tangent line at a. The tangent line to the curve becomes steeper as x approaches a until it becomes a vertical line. Since the slope of a vertical line is undefined, the function is not differentiable in this case. Corner.