Nondifferentiable functions

 1. Absolute Value Function

$$f(x) = |x|$$

This function has a sharp corner (or cusp) at $x = 0$. The slope changes abruptly from negative on the left side of the origin to positive on the right side, making it non-differentiable at $x = 0$.

2. Piecewise Function

A piecewise function can be non-differentiable at the points where its pieces join if there's a sudden change in slope.

$$f(x)=\{\begin{array}{ll}x+1& \text{for }x<0,\\ -x+1& \text{for }x\ge 0.\end{array}$$

This function is non-differentiable at $x = 0$ because the slopes from the left and right do not match.

3. Functions with Vertical Tangents

The function

$$f(x) = \sqrt[3]{x}$$

is differentiable everywhere except at $x = 0$, where the slope becomes infinitely steep (a vertical tangent).

4. Discontinuous Functions

A function that has a jump or gap is not differentiable at the point of discontinuity. For example:

$$f(x) = \begin{cases} 1 & \text{for } x < 0, \\ 2 & \text{for } x \geq 0. \end{cases}$$

This function is non-differentiable at $x = 0$ due to the discontinuity.

In general, a function is not differentiable at points where it is not continuous or where it has abrupt changes in behavior, such as sharp corners or vertical slopes.

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