Give three different ways for a function to fail to be differentiable. Explain why it fails in each case.
The Correct Answer and Explanation is:
A function can fail to be differentiable at a point for several reasons. Here are three common ways:
- Sharp Corners or Cusps
A function is not differentiable at a point where there is a sharp corner or cusp because the left-hand and right-hand derivatives at that point are not equal. A classic example is the absolute value function f(x)=∣x∣f(x) = |x|f(x)=∣x∣ at x=0x = 0x=0. The graph makes a “V” shape at the origin, and although the function is continuous there, the slope from the left (−1-1−1) and the slope from the right (111) are not the same. Since the derivative must be the same from both sides to exist, the function is not differentiable at that point. - Vertical Tangents
A function fails to be differentiable at a point if the tangent line at that point is vertical. This occurs when the slope approaches infinity or does not exist in a finite sense. An example is f(x)=x3f(x) = \sqrt[3]{x}f(x)=3x, whose derivative becomes unbounded as xxx approaches 0. At x=0x = 0x=0, the graph has a vertical tangent. Although the function is continuous and smooth-looking, the steepness of the graph near that point makes it non-differentiable because the derivative tends toward infinity. - Discontinuity
If a function is not continuous at a point, it cannot be differentiable there. Differentiability requires continuity; thus, a discontinuity automatically rules out the existence of a derivative. An example is a step function such as f(x)={1if x<02if x≥0f(x) = \begin{cases}1 & \text{if } x < 0 \\ 2 & \text{if } x \ge 0\end{cases}f(x)={12if x<0if x≥0. At x=0x = 0x=0, the function jumps, creating a break in the graph. Since the limit does not exist at the point of discontinuity, neither does the derivative.
In summary, differentiability demands smoothness and continuity. Sharp turns, infinite slopes, or breaks in the graph each disrupt that smoothness, preventing a derivative from existing.
