Part 4/5:
Understanding the conditions under which a function is non-differentiable is equally important. There are three primary cases where a function may fail to be differentiable:
At Sharp Points: If a graph has a sharp edge or a cusp, like the function ( y = |x| ), the derivative cannot be defined at that point.
Discontinuity: A function may also be non-differentiable if there is a break in the graph (where the function is not continuous).
Infinite Slopes: If at a certain point the slope of the tangent becomes vertical (infinite), the derivative will not exist there.
Example: Absolute Value Function
One classic example is the absolute value function ( y = |x| ). The graph of this function resembles a V-shape, with a sharp point at ( x = 0 ).