Failure of Differentiability

Definition

In the context of calculus, even continuous functions may fail to be differentiable at certain points.

Why It Matters

Real-world systems often have “corners” and “breaks” where smooth calculus fails, and ignoring these points can lead to catastrophic mechanical or algorithmic collapse. Identifying these failures is the difference between a theoretical model and a robust, safe design.

Core Concepts

• Common Failure Cases

1. Corners: The one-sided derivatives exist but are not equal (e.g., $y = |x|$ at $x = 0$).
   • How to read: “The variable y equals the absolute value of x at x equals zero.”
   • Meaning: The graph has a sharp V-turn; left and right slopes disagree, so no single tangent line exists.
2. Cusps: The slopes approach $\infty$ from one side and $-\infty$ from the other.
   • How to read: “The left hand derivative approaches positive infinity, while the right hand derivative approaches negative infinity.”
   • Meaning: The tangent line becomes vertical and reverses direction instantly—a cusp has no well-defined finite slope.
3. Vertical Tangents: The slope of the secant line approaches $\pm\infty$ from both sides (e.g., $y = x^{1/3}$ at $x = 0$).
   • How to read: “The variable y equals x to the one third at x equals zero, and the limit of the quantity f of x plus h minus f of x, all over h, approaches positive or negative infinity.”
   • Meaning: The curve is continuous but too steep for a finite derivative; the tangent is vertical, not a number.
4. Discontinuities: If a function is discontinuous at a point, it cannot be differentiable there.
   • How to read: “If the function f is not continuous at c, then f is not differentiable at c.”
   • Meaning: Differentiability requires the graph to be unbroken; a jump or hole kills the limit definition of the derivative.
5. Wild Oscillation: The function oscillates so rapidly that the limit of the difference quotient does not exist.
   • How to read: “The limit as x approaches c of the quantity f of x minus f of c, divided by the quantity x minus c, does not exist.”
   • Meaning: Secant slopes swing wildly without settling on one value, so the instantaneous rate of change is undefined.

Geometrically, these cases correspond to points where the graph does not have a unique, non-vertical tangent line.

Connected Concepts

• Failure of Limits to Exist
• Differentiability Implies Continuity