Problem 3 - A Primer of Real Analysis

Question

Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by $$ f(x)=\left\\{\begin{array}{ll} x, & \text { if } x<0 \\ x^{2}, & \text { if } x \geq 0 \end{array}\right. $$ Is \(f\) differentiable at \(0 ?\)

Step-by-Step Solution

Answer

No, \( f \) is not differentiable at 0 because the left and right limits are different.

Step 1: Define Differentiability

A function \( f \) is differentiable at a point \( c \) if the following limit exists: \[ \lim_{{h \to 0}} \frac{{f(c+h) - f(c)}}{h}\]In this case, you must consider \( c = 0 \).Additionally, the left-hand limit and the right-hand limit at \( c \) must coincide.

Step 2: Evaluate Function at Zero

Evaluate \( f \) at \( x = 0 \). Since \( 0 \geq 0 \), use the second definition: \( f(0) = 0^2 = 0 \).

Step 3: Calculate Right-Hand Limit

For \( h > 0 \), evaluate the right-hand limit:\[ \lim_{{h \to 0^+}} \frac{{f(h) - f(0)}}{h} = \lim_{{h \to 0^+}} \frac{{h^2 - 0}}{h} = \lim_{{h \to 0^+}} h = 0.\]

Step 4: Calculate Left-Hand Limit

For \( h < 0 \), evaluate the left-hand limit:\[ \lim_{{h \to 0^-}} \frac{{f(h) - f(0)}}{h} = \lim_{{h \to 0^-}} \frac{{h - 0}}{h} = \lim_{{h \to 0^-}} 1 = 1.\]

Step 5: Conclude Differentiability

Since the right-hand limit (0) and the left-hand limit (1) at \( x = 0 \) are not equal, the limit \[ \lim_{{h \to 0}} \frac{{f(h) - f(0)}}{h}\]does not exist. Therefore, \( f \) is not differentiable at \( x = 0 \).

Key Concepts

Piecewise function

A piecewise function is defined by different expressions depending on the input value's domain. This means that the function's expression can change based on the value of the input, allowing it to capture different behaviors over different intervals.
In the case of our function, we see:

• For values of \( x \) less than 0, the function \( f(x) = x \) applies. Here, the output is exactly the same as the input value.
• For values of \( x \) greater than or equal to 0, the expression \( f(x) = x^2 \) applies, which means the output is the square of the input value.

This structure allows us to analyze the function's differentiability at specific points like 0, where the expression switches from one form to another.

Right-hand limit

A right-hand limit focuses on approaching a particular point from the right, or from values greater than your point of interest. This is important when examining limits as you need to confirm behavior from both directions.
To calculate the right-hand limit for a point \( c \), we examine \( \lim_{{h o 0^+}} \frac{{f(c+h) - f(c)}}{h} \). In our function, evaluating this at \( x=0 \) with \( f(x) = x^2 \) yields:

• \( f(h) = h^2 \) when approaching 0 from the positive side.
• The calculation simplifies to \( \lim_{{h \to 0^+}} h = 0 \).

Thus, the right-hand limit at the point 0 is 0. This means that from the right, the slope of the tangent to the function is 0.

Left-hand limit

The left-hand limit is concerned with the behavior of a function as it approaches a certain point from the left, i.e., from values less than the point. Looking at both sides is crucial to verify differentiability.
For the left-hand limit at a point \( c \), we compute \( \lim_{{h \to 0^-}} \frac{{f(c+h) - f(c)}}{h} \). In examining our function at \( x=0 \) with \( f(x) = x \) as we approach from the left:

• We consider \( f(h) = h \) for values approaching 0 from the left side.
• This simplifies to \( \lim_{{h \to 0^-}} 1 = 1 \), indicating a constant slope of 1.

Thus, the left-hand limit at 0 shows that the slope is 1 as the function approaches from the left.

Existence of limit

The existence of a limit at a point is determined by confirming that the function behaves consistently from both sides of the point. This means both the right-hand limit and the left-hand limit must be identical for the limit to truly exist at that point.
For a function to be differentiable at a point, this consistent behavior is essential because it translates to the derivative—essentially the function's slope—being the same from both directions.

• In our exercise, calculating limits at \( x=0 \), we see that the right-hand limit is 0, and the left-hand limit is 1.
• Since these two values differ, the overall limit at \( x=0 \) does not exist, which tells us the function is not differentiable at that point.

Understanding the existence of limits helps explain why our function behaves differently in each segment and indicates that there is a "kink" or point of discontinuity in the derivative at the transition point between pieces.