Reasons Why Functions May Not be Differentiable | Understanding Discontinuity, Corners, Vertical Tangents, and More

Reasons for a function to not be differentiable

There are several reasons why a function may not be differentiable at a certain point or over a given interval:

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There are several reasons why a function may not be differentiable at a certain point or over a given interval:

1. Discontinuity: If a function has a jump, infinite, or removable discontinuity at a certain point, it will not be differentiable at that point. A jump discontinuity occurs when the function has a sudden jump in its values. An infinite discontinuity occurs when the function approaches positive or negative infinity at a particular point. A removable discontinuity refers to a hole in the graph where the function could be made continuous by defining its value at that point.

2. Corner or cusp: At points where a function has a sharp change in direction, like a corner or cusp, it will not be differentiable. The derivative represents the slope of the function, and at such points, the slope can’t be defined because there is no unique tangent line.

3. Vertical tangent: If the slope of a function becomes infinite at a particular point, it will not be differentiable at that point. This occurs when the function has a vertical tangent line, which indicates an extremely steep slope.

4. Non-smoothness: Functions that have sharp corners, cusps, or vertical tangents are considered non-smooth, and they lack differentiability at these specific points. Additionally, functions that have kinks or sharp bends in their graph may also not be differentiable at those points.

5. Oscillatory behavior: Some functions may oscillate rapidly or chaotically, preventing the existence of a consistent tangent line and hence differentiability. Examples of such functions include the Dirichlet function or the Weierstrass function.

6. Infinite rate of change: If a function has an infinite rate of change at a point, it will not be differentiable at that point. This can happen when the function contains singularities or points where the function rapidly increases or decreases without bound.

It’s important to note that a function may still be differentiable over most of its domain, even if it lacks differentiability at isolated points or intervals.

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