How to Determine Differentiability of a Piecewise Function at a Point

1. What We're Solving:

We have a piecewise function that behaves differently for rational and irrational numbers:

• f(x) = x/2 when x is rational (x ∈ ℚ)
• f(x) = x when x is irrational (x ∉ ℚ)

We need to determine if this function is differentiable at x = 0. This is a beautiful example of how continuity and differentiability can behave in surprising ways!

2. The Approach:

To check differentiability at x = 0, we need to see if the derivative limit exists: $$\lim_{h \to 0} \frac{f(h) - f(0)}{h}$$

Since our function is piecewise, we'll need to be clever about how we approach this limit. The key insight is that we can approach 0 through different types of numbers (rationals vs. irrationals) and see if we get the same limit!

3. Step-by-Step Solution:

Step 1: Find f(0) Since 0 is rational, f(0) = 0/2 = 0. Great!

Step 2: Set up the difference quotient We need: $\lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{f(h) - 0}{h} = \lim_{h \to 0} \frac{f(h)}{h}$

Step 3: Consider what happens as h approaches 0 Here's where it gets interesting! As h approaches 0, we have:

• When h is rational: $\frac{f(h)}{h} = \frac{h/2}{h} = \frac{1}{2}$
• When h is irrational: $\frac{f(h)}{h} = \frac{h}{h} = 1$

Step 4: Use the density of rationals and irrationals This is the crucial step! Both rational and irrational numbers are dense in the real numbers, meaning:

• Any neighborhood of 0 contains infinitely many rational numbers
• Any neighborhood of 0 contains infinitely many irrational numbers

Step 5: Analyze the limit Since we can find sequences approaching 0 where the difference quotient equals 1/2 (rational sequences) and other sequences where it equals 1 (irrational sequences), the limit doesn't exist!

For the limit to exist, ALL sequences approaching 0 must give the same value for the difference quotient.

4. The Answer:

The function f(x) is NOT differentiable at x = 0.

The derivative limit doesn't exist because:

• Approaching through rational numbers gives us a derivative of 1/2
• Approaching through irrational numbers gives us a derivative of 1
• Since 1/2 ≠ 1, the limit doesn't exist

5. Memory Tip:

Think of this as a "split personality" function at x = 0! Even though the function is continuous at 0 (both pieces give f(0) = 0), it "can't decide" what slope it wants to have. Rationals pull it toward slope 1/2, while irrationals pull it toward slope 1. This tug-of-war means no single derivative exists!

This is a perfect example of why differentiability is stronger than continuity - a function can be continuous but still fail to be differentiable when it has conflicting "directional behaviors."