# OEF differentiability --- Introduction ---

This module actually gathers 6 exercises on the differentiability (definition and fundamental properties) of functions of one real variable.

### abs

What is the differentiability of the function f(x) =  over the interval [-10,10]?

### Absolute order

Let : -> be the function defined by (x) = . What is the order of differentiability of  ?

Instructions/Examples.

• Type 3 if is differentiable to order 3 but not to order 4.
• Type 0 if is continuous but not differentiable.
• Type -1 if is not continuous.
• Type if is differentiable to any order.

### Continuity of derivative

Let : -> be a continuous function. If the derivative (x) exists for any point x  , is the derivative function : -> always continuous?

### Continuity of derivative II

Let : -> be a continuous function. Suppose that the derivative (x) exists for any point x.

If furthermore , is the derivative function : -> always continuous?

### Non-differentiable inverse

The function : -> defined by

(x) =

is bijective, but there is a point    such that the inverse function -1(x) is not differentiable on . Find .

### Sided order

Let : -> be the function defined by

 (x) = si x < ; si x .

What is the order of differentiability of  ?

Instructions/Examples.

• Type 3 if is differentiable to order 3 but not to order 4.
• Type 0 if is continuous but not differentiable.
• Type -1 if is not continuous.
• Type if is differentiable to any order.

Other exercises on: differentiability   continuity   calculus