Determining if a piecewise-defined function is differentiable at the origin requires a systematic approach that bridges the gap between algebraic manipulation and the fundamental geometric definition of a derivative. Consider this: unlike standard functions defined by a single expression, piecewise functions introduce a potential "break" or "corner" precisely at the boundary points—in this case, at x = 0. That said, to establish differentiability at the origin, one must verify two distinct but interconnected conditions: the function must be continuous at that point, and the left-hand and right-hand derivatives must exist and be equal. This article provides a complete walkthrough to navigating this process, complete with the theoretical framework, a step-by-step methodology, and illustrative examples to solidify your understanding.
People argue about this. Here's where I land on it.
The Theoretical Foundation: Continuity and Smoothness
Before diving into the mechanics of differentiation, it is crucial to internalize a fundamental theorem of calculus: differentiability implies continuity. Worth adding: if a function f(x) is differentiable at x = 0, it is automatically continuous there. Conversely, if the function is discontinuous at the origin, it cannot be differentiable. This logical dependency dictates the order of operations; checking continuity is the necessary "gatekeeper" step before attempting to compute derivatives.
For a piecewise function defined as: $ f(x) = \begin{cases} g(x) & \text{if } x < 0 \ h(x) & \text{if } x \geq 0 \end{cases} $ continuity at the origin demands three specific criteria be met simultaneously:
- The left-hand limit exists: $\lim_{x \to 0^-} g(x) = L$. In practice, 2. The right-hand limit exists: $\lim_{x \to 0^+} h(x) = L$. Practically speaking, 3. The function value is defined and matches the limit: $f(0) = h(0) = L$.
If any of these fail, the investigation ends immediately—the function is not differentiable at the origin. Only when the graph connects smoothly at (0, f(0)) do we proceed to analyze the "smoothness" of that connection But it adds up..
The Definition of the Derivative at a Boundary
The derivative of f at x = 0 is defined by the limit of the difference quotient: $ f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} $ Because the definition of f changes depending on the sign of h, this single two-sided limit must be split into two separate one-sided limits. The derivative exists if and only if both the left-hand derivative and the right-hand derivative exist finitely and are equal.
- Left-hand derivative ($f'_-(0)$): Uses the definition for $x < 0$ (usually $g(x)$). $ f'-(0) = \lim{h \to 0^-} \frac{g(h) - f(0)}{h} $
- Right-hand derivative ($f'_+(0)$): Uses the definition for $x \geq 0$ (usually $h(x)$). $ f'+(0) = \lim{h \to 0^+} \frac{h(h) - f(0)}{h} $
The Verdict: The function is differentiable at the origin if and only if $f'-(0) = f'+(0)$. If these values differ, the graph has a corner or cusp at the origin. If either limit is infinite or does not exist, the function has a vertical tangent or oscillatory behavior, respectively, and is not differentiable in the standard sense Nothing fancy..
Step-by-Step Procedure for Analysis
To ensure a rigorous and error-free evaluation, follow this structured workflow every time you encounter a piecewise function at a boundary point.
Step 1: Verify Continuity at $x = 0$
Calculate $\lim_{x \to 0^-} f(x)$, $\lim_{x \to 0^+} f(x)$, and $f(0)$.
- If $\lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x)$, stop. Not differentiable (jump discontinuity).
- If limits are equal but $\neq f(0)$, stop. Not differentiable (removable discontinuity).
- If all three match, proceed to Step 2.
Step 2: Compute the Left-Hand Derivative
Set up the difference quotient using the "left-side" rule ($x < 0$). $ f'-(0) = \lim{h \to 0^-} \frac{f(h) - f(0)}{h} $ Simplify the algebraic expression carefully. Pay close attention to the sign of $h$ (negative) when simplifying absolute values or square roots. Evaluate the limit.
Step 3: Compute the Right-Hand Derivative
Set up the difference quotient using the "right-side" rule ($x \geq 0$). $ f'+(0) = \lim{h \to 0^+} \frac{f(h) - f(0)}{h} $ Simplify and evaluate. Note that here $h > 0$.
Step 4: Compare and Conclude
- If $f'-(0) = f'+(0) = m$ (a finite number), $f$ is differentiable at 0 and $f'(0) = m$.
- If $f'-(0) \neq f'+(0)$, $f$ is not differentiable at 0 (corner/cusp).
- If either limit is $\pm \infty$ or DNE, $f$ is not differentiable at 0.
Worked Examples: From Standard to Subtle
Example 1: The Classic Absolute Value (Corner)
Determine differentiability of $f(x) = |x|$ at the origin. Piecewise form: $ f(x) = \begin{cases} -x & x < 0 \ x & x \geq 0 \end{cases} $
Step 1: Continuity $\lim_{x \to 0^-} -x = 0$, $\lim_{x \to 0^+} x = 0$, $f(0) = 0$. Continuous.
Step 2: Left-Hand Derivative $ f'-(0) = \lim{h \to 0^-} \frac{-h - 0}{h} = \lim_{h \to 0^-} -1 = -1 $
Step 3: Right-Hand Derivative $ f'+(0) = \lim{h \to 0^+} \frac{h - 0}{h} = \lim_{h \to 0^+} 1 = 1 $
Step 4: Conclusion $-1 \neq 1$. The function has a sharp corner at the origin. Not differentiable.
Example 2: Smooth Transition (Differentiable)
Determine differentiability of $f(x) = \begin{cases} x^2 & x < 0 \ x^3 & x \geq 0 \end{cases}$ at the origin.
Step 1: Continuity $\lim_{x \to 0^-} x^2 = 0$, $\lim_{x \to 0^+} x^3 = 0$, $f(0) = 0$. Continuous.
Step 2: Left-Hand Derivative $ f'-(0) = \lim{h \to 0^-} \frac{h^2 - 0}{h} = \lim_{h \to
