What Defines Whether a Function is Differentiable?

Fundamental Definition

A function \(f(x)\) is differentiable at a point \(x = a\) if the derivative exists at that point. Mathematically, this means the following limit exists:

\(f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\)

If this limit exists and is finite, we say \(f\) is differentiable at \(x = a\).

Key Conditions for Differentiability

1. Continuity

Differentiability implies continuity: If a function is differentiable at a point, it MUST be continuous there.

\(\text{If } f \text{ is differentiable at } a \Rightarrow \lim_{x \to a} f(x) = f(a)\)

However, continuity does NOT guarantee differentiability.

2. Smoothness

The function must have a well-defined, non-vertical tangent line at the point. The graph should not have:

• • Corners or cusps
• • Vertical tangents
• • Discontinuities
• • Sharp turns

3. Left and Right Derivatives

For differentiability at a point, the left-hand derivative and right-hand derivative must exist and be equal:

\(\lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h} = \lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h}\)

Examples of Differentiable Functions

\(f(x) = x^2\)    (Differentiable everywhere)
\(f(x) = \sin(x)\)    (Differentiable everywhere)
\(f(x) = e^x\)    (Differentiable everywhere)

These functions are smooth and continuous with no sharp corners or breaks.

Common Cases of Non-Differentiability

1. Corners (Absolute Value Function)

\(f(x) = |x|\) at \(x = 0\)

Left derivative = -1, Right derivative = +1 ⇒ Not differentiable at x = 0

2. Cusps

\(f(x) = x^{2/3}\) at \(x = 0\)

Vertical tangent ⇒ Derivative approaches ±∞

3. Discontinuities

\(f(x) = \frac{1}{x}\) at \(x = 0\)

Not continuous ⇒ Not differentiable

4. Vertical Tangents

\(f(x) = \sqrt[3]{x}\) at \(x = 0\)

Derivative becomes infinite

Higher-Order Differentiability

Continuously Differentiable (C¹)

A function is continuously differentiable if its derivative exists and is continuous.

Smooth Functions (C∞)

A function is smooth if it has derivatives of all orders that are continuous.

Examples: Polynomials, exponential functions, trigonometric functions

Summary: Differentiability Checklist

A function \(f(x)\) is differentiable at \(x = a\) if:

1. Continuity: \(f\) is continuous at \(a\)
2. Limit Existence: The derivative limit exists and is finite
3. Left-Right Equality: Left-hand derivative = Right-hand derivative
4. Smoothness: No corners, cusps, or vertical tangents

When analyzing bifurcations in dynamical systems, we typically work with smooth (C∞) functions, ensuring differentiability along bifurcating branches except at degenerate bifurcation points.