Show that the function |x| is not differentiable at all points?
Answers
Case 1 :
Firstly check the continuity of that function.
The function is not differentiable at any point if the function is discontinuous at that point..
Because the curve doesn't exist at the point of discontinuity, so not differentiable at that point.
Case 2 :
In some specific cases, a function may not be differentiable at the point where the function is continuous.
Such points are breaking point.
** Breaking points are such points where there is not possible to draw a tangent to the curve at that point .
For an example y = |x| is a function which is continuos at (0,0) but not differentiable.
For the function y = |x| , (0,0) is a breaking point.
** And if you find the left hand limit, L f ' (P) and right hand limit, R f ' (P) of a function f(x) at point P,
The function will be differentiable if
(i) L f ' (P) and R f ' (P) are defined.
(ii) L f ' (P) = R f ' (P) ..
Thus, You can also check differentiability of a function using the concept of limit.
