Why Functions Fail to be Differentiable | Understanding Discontinuity, Sharp Corners, and Vertical Tangents

3 ways in which a function to not be differentiable at a point

A function can fail to be differentiable at a specific point for a variety of reasons

A function can fail to be differentiable at a specific point for a variety of reasons. Here are three common ways:

1. Discontinuity: A function is not differentiable at a point if it is discontinuous at that point. Discontinuity occurs when there is a sudden jump, hole, or vertical asymptote in the graph of the function. At points of discontinuity, the concept of a derivative does not apply as there is no unique tangent line that can be drawn.

2. Sharp corner or cusp: A function is not differentiable at a point if it has a sharp corner or cusp at that point. A sharp corner occurs when the slopes of the tangent lines approaching from either side of the point are different, leading to a discontinuity in the derivative. Similarly, a cusp occurs when the tangent lines approach the point but do not have a well-defined limiting slope.

3. Vertical tangent: A function is not differentiable at a point if it has a vertical tangent line at that point. A vertical tangent occurs when the slope of the function approaches infinity or negative infinity. In such cases, the derivative does not exist since the function becomes infinitely steep, making it impossible to determine a unique tangent line.

It is important to note that the absence of differentiability at a point does not imply that the function is not differentiable elsewhere. A function may still be differentiable at other points even if it fails to be differentiable at a specific point.

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