## Are differentiable functions dense?

The collection of continuous nowhere differentiable functions is dense in the Banach space X = C([0,1]) of continuous functions on [0,1] with the supremum norm.

**What Is the Meaning of Nowhere differentiable?**

A function f:S⊆R→R f : S ⊆ ℝ → ℝ is said to be nowhere differentiable. if it is not differentiable at any point in the domain S of f . It is easy to produce examples of nowhere differentiable functions.

### What does it mean when a function isn’t 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.

**Which functions are differentiable everywhere?**

Sines and cosines and exponents are differentiable everywhere but tangents and secants are singular at certain values.

## Is every differentiable function Lipschitz?

In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is locally bounded as well.

**What kinds of functions are not 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.

### What does it mean if a graph is differentiable?

Simply put, differentiable means the derivative exists at every point in its domain. Consequently, the only way for the derivative to exist is if the function also exists (i.e., is continuous) on its domain. Thus, a differentiable function is also a continuous function.

**Does f satisfy a Lipschitz condition on D?**

Theorem 1 Suppose f(t,y) is defined on a convex set D in R2. If a constant L 0 exists with ∂f ∂y t, y ≤ L, for all t, y in D, then f satisfies a Lipschitz condition on D in the variable y with Lipschitz constant L.

## What is Lipschitz?

A function f is called L-Lipschitz over a set S with respect to a norm ‖·‖ if for all u,w∈S we have: Some people will equivalently say f is Lipschitz continuous with Lipschitz constant L. Intuitively, L is a measure of how fast the function can change.