Understanding Differentiability in Mathematics | Common Scenarios Where Functions Fail to be Differentiable

When is a function not differentiable?

A function is not differentiable at a particular point if it fails to meet one or more conditions necessary for differentiation at that point

A function is not differentiable at a particular point if it fails to meet one or more conditions necessary for differentiation at that point. There are several scenarios where a function may not be differentiable:

1. Corner or Cusp: If a function has a corner or a cusp at a particular point, it is not differentiable at that point. This occurs when the function abruptly changes direction, resulting in a sharp turn or a point of discontinuity.

2. Vertical Tangent: If a function has a vertical tangent at a particular point, it is not differentiable at that point. This means that the slope of the function becomes undefined or infinite, resembling a vertical line.

3. Discontinuity: If a function has a discontinuity at a particular point, it is generally not differentiable at that point. Discontinuities can be categorized as removable (where the function can be modified to be continuous by redefining a single point) or non-removable (where the function cannot be modified to be continuous at that point).

4. Jump Discontinuity: A jump discontinuity occurs when the function has a sudden change in its value at a particular point. In such cases, the function is not differentiable at that point.

5. Oscillation: If a function oscillates rapidly or exhibits “wiggles” at a particular point, it may not be differentiable at that point. This often happens in functions that have rapid oscillations or irregular behavior.

6. Infinite Slope: If a function has an infinite slope at a particular point, it is not differentiable at that point. This occurs when the tangent line of the function becomes vertical, indicating an undefined slope.

It is important to note that a function can still be differentiable at most points even if it is not differentiable at a handful of isolated points.

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