Understanding Non-Differentiability in Functions: Discontinuities, Sharp Corners, and Undefined Derivatives

A function is not differentiable at places where there is…

A function is not differentiable at places where there is a discontinuity or a sharp corner, or where the derivative is undefined

A function is not differentiable at places where there is a discontinuity or a sharp corner, or where the derivative is undefined.

Let’s break down these scenarios in further detail:

1. Discontinuity: Discontinuities occur when there is a jump or a break in the function. There are three types of discontinuities: removable, jump, and essential discontinuities. A removable discontinuity can potentially be fixed by redefining the function value at that specific point. A jump discontinuity occurs when the function approaches different values from the left and the right sides of a specific point. An essential discontinuity happens when the function does not approach any finite value at a specific point.

At such points of discontinuity, the derivative does not exist, and hence the function is not differentiable there.

2. Sharp corner: A sharp corner occurs when the function changes its direction abruptly, forming a sharp angle at a specific point. At these points, the derivative does not exist since the tangent line cannot be defined due to the abrupt change in direction.

3. Undefined derivative: There are certain situations where the derivative cannot be defined. For example, when a function has a vertical tangent line or vertical asymptote, the slope of the tangent line at that point becomes infinite. Similarly, when a function has an infinite oscillation or does not approach any definite value, the derivative is undefined.

In summary, a function is not differentiable at places where there is a discontinuity, sharp corner, or an undefined derivative. These points typically indicate an abrupt change or irregularity in the function’s behavior.

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