Problem 59 - Single Variable Calculus

Question

Suppose that \(g(x)=|x-a| f(x)\), where \(f\) is a continuous function and \(f(a) \neq 0\). Show that \(g\) is continuous at \(a\) but not differentiable at \(\underline{a}\).

Step-by-Step Solution

Answer

In conclusion, the function \(g(x) = |x-a| f(x)\) is continuous at \(a\) since the right and left limits both equal to 0 as \(x\) approaches \(a\). However, it is not differentiable at \(a\) because the derivative of \(|x-a|\) is not defined at \(x=a\).

Step 1: Check the continuity of g(x) at a

To check the continuity of \(g(x)\) at \(a\), we need to find the limit as \(x\) approaches \(a\). This can be done by finding the left and right limits and comparing them. The left limit is given by: \[\lim_{x \to a^-} g(x) = \lim_{x \to a^-} |x-a| f(x)\] Since \(xa\), \(|x-a| = x-a\). So, we have: \[\lim_{x \to a^+} g(x) = \lim_{x \to a^+} (x-a) f(x)\] As before, we substitute \(a\) for \(x\) and get: \[\lim_{x \to a^+} g(x) = (a-a) f(a) = 0\] Since the left and right limits are equal, the limit exists and \(g(x)\) is continuous at \(a\).

Step 2: Check the differentiability of g(x) at a

Now, let's check for differentiability by calculating the derivative of \(g(x)\) using the product rule: \[\frac{d}{dx} g(x) = \frac{d}{dx} (|x-a| f(x)) = (|x-a| f(x))'\] Using the product rule: \[(|x-a| f(x))' = |x-a|' f(x) + (|x-a|) (f(x))'\] The derivative of \(|x-a|\) is not defined at \(x=a\) since it has a sharp corner. Therefore, the derivative of \(g(x)\) does not exist at \(a\). In conclusion, the given function \(g(x) = |x-a| f(x)\) is continuous at \(a\) but not differentiable at \(a\).

Key Concepts

Continuous Functions

A continuous function is one that can be drawn without lifting your pencil from the paper. This intuitive definition actually comes down to the mathematical concept that a function is continuous at a point if the limit of the function as it approaches this point from both directions equals the function's value at the point.

In the problem, we explore the function \(g(x) = |x - a| f(x)\), where \(f(x)\) is known to be continuous. To establish the continuity of \(g(x)\) at \(x = a\), we calculate the left-hand and right-hand limits as \(x\) approaches \(a\).

• Left limit: \(\lim_{x \to a^-} |x - a| f(x) = 0\)
• Right limit: \(\lim_{x \to a^+} |x - a| f(x) = 0\)

Since these two limits are equal, the overall limit exists and equals \(g(a)\), implying that \(g(x)\) is continuous at \(x = a\). This exercise demonstrates the use of limits in establishing the continuity of composite functions, even when they involve absolute values.

Differentiability

Differentiability concerns the presence of a derivative, which in intuitive terms means the function has a definite slope or rate of change at a point. For a function to be differentiable at a specific point, it must be smooth there—no sharp corners or cusps.

In this exercise, we examine whether \(g(x) = |x-a| f(x)\) is differentiable at \(x = a\). We use the product rule to attempt to differentiate \(g(x)\), but quickly face a roadblock: the absolute value \(|x-a|\) introduces a sharp corner at \(x = a\), making the derivative undefined.
The issue boils down to:

• The function \(|x-a|\) is not smooth at \(x=a\), causing its derivative to be undefined exactly at that point.
• As a result, when calculating \(|x-a|' f(x)\), there's no proper slope at \(x = a\), reminding us that differentiable functions must be continuous yet not every continuous function is differentiable.

This example is a classic case where continuity doesn't guarantee differentiability due to a sharp change in direction.

Limits

Limits are the foundational building blocks of calculus. They help us understand behaviors of functions around given points and are crucial in defining both continuity and differentiability. In our problem, the function \(g(x) = |x - a| f(x)\) employs limits to investigate its behavior as \(x\) approaches \(a\).

By dividing the examination into left-hand and right-hand limits around the point \(x = a\), we ascertain the continuity of \(g(x)\) by showing both limits reaching the same value.

• Left-hand limit: \(\lim_{x \to a^-} |x-a| f(x)\) intuitively reflects the function approaching 0 from the left.
• Right-hand limit: \(\lim_{x \to a^+} |x-a| f(x)\) similarly approaches 0 from the right.

This study of limits showcases the subtlety required in handling functions with absolute values, especially when aspiring to establish the broader properties like continuity and differentiability. Understanding limits and their application allows us a window into more complex function behavior and how foundational concepts interrelate in calculus.