abs

What is the differentiability of the function f(x) = $val10 over the interval [-10,10]?

Absolute order

Let $val6: $val7 -> $val7 be the function defined by $val6(x) = $val19. What is the order of differentiability of $val6 ?

Instructions/Examples.


Continuity of derivative

Let $val6: $val7 -> $val7 be a continuous function. If the derivative $val8(x) exists for any point x  $val7, is the derivative function $val8: $val7 -> $val7 always continuous?

Continuity of derivative II

Let $val8: $val9 -> $val9 be a continuous function. Suppose that the derivative $val10(x) exists for any point x$val11$val9.

If furthermore $val16, is the derivative function $val10: $val9 -> $val9 always continuous?


Non-differentiable inverse

The function $val6: $val7 -> $val7 defined by

$val6(x) = $val15

is bijective, but there is a point $val8  $val7 such that the inverse function $val6-1(x) is not differentiable on $val8. Find $val8.


Sided order

Let $val6: $val7 -> $val7 be the function defined by

$val6(x) = $val25      si x < $val9 ;
$val26 si x $val9 .

What is the order of differentiability of $val6 ?

Instructions/Examples.