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.

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.

Other exercises on:
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