Non-differentiable

In calculus, a function is considered **non-differentiable** at a point if its derivative does not exist at that point. This typically occurs due to several factors: a sharp corner or cusp, a vertical tangent, a discontinuity (a jump, a hole, or an asymptote), or oscillating behavior. Essentially, the function's rate of change is undefined or changes abruptly at that specific location, making a unique tangent line impossible to define. Examining the limit of the difference quotient is crucial to identify these points and understand the function's behavior near them.

Non-differentiable meaning with examples

• The absolute value function, f(x) = |x|, is non-differentiable at x = 0 because of the sharp corner. Approaching 0 from the left gives a slope of -1, while approaching from the right gives a slope of +1. The lack of a consistent slope prevents a well-defined derivative, a clear example showcasing the concept. The derivative is not continuous.
• Consider the function representing the path of a ball bouncing perfectly on the floor, creating sharp corners, that are not smooth. The instantaneous velocity changes direction abruptly each time the ball hits. The graph of this path is not differentiable at any point of impact, illustrating the concept in the physical world and how a non-smooth path does not have a derivative
• Functions with discontinuities, such as those with jumps or removable discontinuities, are non-differentiable at the points of discontinuity. For instance, a step function will fail the test of differentiability at the point where the step occurs. The derivative is not defined at the point of a gap and that is why it is a good example, as differentiability requires continuity.
• The Weierstrass function, a classic example, is continuous everywhere but non-differentiable anywhere. This pathological function underscores that continuity is necessary but not sufficient for differentiability. This function showcases how complex a non-differentiable function can be and that they don't need obvious features, such as the previous examples, to exhibit this trait.