Question

How do you find the non-differentiable points for a function?

Answer 1

Hint: In this question we have used the properties of limits and derivatives to find out at which points a function is not differentiable. We will list all the possible conditions where a point for a function is not differentiable.

Complete step-by-step solution:
A function is not differentiable on the following conditions:
$a)$ A function is not differentiable if it is discontinuous. The property of a continuous function is that it is in the form of curved lines.
We should first check the continuity of the function, because there is no curve at that point than it is not differentiable
$b)$ In the above-mentioned property, we learn that we cannot differentiate a function where it is discontinuous, but a function can still be discontinuous if it is continuous, in some specific conditions, the breaking point might occur.
A breaking point is defined as a point where the tangent to the curve cannot be formed. To easily recognize such a point on a graph any corner point or cusp should be looked for. Using the property of limit, we can write it as:
$\mathop {\lim }\limits_{h \to {0^ - }} \dfrac{{f(a + h) - f(a)}}{h} \ne \mathop {\lim }\limits_{h \to {0^ + }} \dfrac{{f(a + h) - f(a)}}{h}$
An example to this case is the function $y = |x|$ which is continuous at $(0,0)$ but not differentiable at the same point, because it is a breaking point.
$c)$ A point is still not differentiable if the tangent to the curve at that point is a vertical line because the derivative of that point reaches to infinity which is not a number. Using limits, it can be written as:
$\mathop {\lim }\limits_{x \to {0^ - }} |f'(x)| = \infty $ or $\mathop {\lim }\limits_{x \to {0^ + }} |f'(x)| = \infty $.

Note: It is to be remembered that limits are the stone on top of which the concepts of integration and derivative are based.
The application of using differentiation is to find the rate of change in quantities.