Common Reasons Why a Function May Not Be Differentiable | Discontinuity, Corners & Cusps, Vertical Tangents, Sharp Points, and Jump Discontinuity

qualities that make a function not differentiable

There are a few qualities that can make a function not differentiable at a specific point or in a certain interval

There are a few qualities that can make a function not differentiable at a specific point or in a certain interval. Here are some of the most common reasons:

1. Discontinuity: A function is not differentiable if it has a point of discontinuity in the vicinity. Discontinuity refers to a sudden jump, hole, or vertical asymptote in the graph of the function. At these points, the function does not have a well-defined slope, and therefore, it is not differentiable.

2. Corner or cusp: If the graph of a function has a sharp corner (also known as a corner point) or a cusp, the function is not differentiable at that point. At these points, the tangent line does not exist and the concept of a slope is not well-defined.

3. Vertical tangent: A function is not differentiable if it has a vertical tangent line at a point. This occurs when the slope of the tangent line becomes infinite, meaning there is a sharp vertical change in the graph at that point.

4. Sharp point: In some cases, a function may have a sharp point or a sharp peak in the graph where the derivative does not exist. This typically happens when the function has a sharp change in direction or curvature at that point.

5. Jump discontinuity: A function is not differentiable if it has a jump discontinuity, which is a sudden change in the function’s value between two points. At the point of the jump, the function does not have a well-defined tangent line and, therefore, is not differentiable.

It is important to note that while a function may not be differentiable at certain points or in certain intervals, it can still be differentiable elsewhere. Differentiability is a local property, meaning it is concerned with the behavior of the function in the immediate vicinity of a specific point or interval.

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