Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The function \(f(x)=x+|x|\) is continuous at \(x=0\) but not differentiable at \(x=0\). Explain, using the de nitions of continuity at a point and differentiability at a point.

Step-by-Step Solution

Answer

The function \(f(x) = x + |x|\) is continuous at \(x = 0\) because the left and right limits equal the function's value at 0. However, it's not differentiable at \(x = 0\) because the left and right derivatives at 0 don't match.

Step 1: Understanding and evaluating continuity.

Continuity of a function at a point means that the function value approaches the same number from both sides as we makes x values close to that point. For continuity of a function at point \(x = a\), the limit, \(\lim_ {x\rightarrow a^+} f(x)\) should equal to \(\lim_ {x\rightarrow a^-} f(x)\) and to \(f(a)\). Evaluate the function given: \(f(x) = x + |x|\). When \(x < 0\), \(f(x) = x - x = 0\). When \(x > 0\), \(f(x) = x + x = 2x\). At \(x = 0\), \(f(x) = 0 + 0 = 0\). So, \(\lim_ {x\rightarrow 0^-} f(x) = 0 = f(0)\), \(\lim_ {x\rightarrow 0^+} f(x) = 0 \), proves that f(x) is continuous at \(x = 0\).

Step 2: Understanding and evaluating differentiability.

Differentiability of a function at a point means that the function has a derivative at that point, i.e., the slope of the tangent line to the function exists and is well defined. For differentiability, \(\lim_ {x\rightarrow a^+} f'(x)\) should equal to \(\lim_ {x\rightarrow a^-} f'(x)\). Analyze \(f(x)\) when applying this. When \(x < 0\), \(f'(x) = 1 - 1 = 0\). When \(x > 0\), \(f'(x) = 1 + 1 = 2\). The limits on either sides of 0 don't match, meaning \(f(x)\) lacks a derivative at \(x = 0\), so it's not differentiable at \(x = 0\).

Step 3: Summary of findings

To summarize, \(f(x) = x + |x|\) is continuous at \(x = 0\) due to the match between the value of the function and the limits approaching from either side. However, it's not differentiable at that point, as the derivatives on either side of 0 don't match. Thus, even though the function is free of any breaks, holes, or jumps at \(x = 0\), making it continuous, the function fails to have a unique, well-defined derivative at that point, rendering it non-differentiable.

Key Concepts

Continuity

Continuity of a function at a point is a fundamental concept in calculus. A function is said to be continuous at a particular point if, as we approach the point from either side, the function's value remains consistent with the value of the function exactly at that point.

For a function to be continuous at a point \(x = a\), several conditions must be satisfied:

• The limit of the function as \(x\) approaches the point from the left (\(\lim_{x\rightarrow a^-} f(x)\)) must equal the limit of the function as \(x\) approaches from the right (\(\lim_{x\rightarrow a^+} f(x)\)).
• Both of these limits should be equal to the actual value of the function at that point (\(f(a)\)).

In the given function, \(f(x) = x + |x|\), analyzing around \(x = 0\) shows that both limits from the left and right as well as the value at \(x = 0\) itself, converge to 0. This makes the function continuous at that point.

Differentiability

Differentiability is another crucial concept in calculus that refers to whether a function has a derivative at a specific point.

If a function is differentiable at a point \(x = a\), it means there exists a well-defined tangent to the curve on the graph of the function at that point — in other words, the slope of the function at \(x = a\) is unique. To determine differentiability:

• The derivative of the function from the left (\(\lim_{x\rightarrow a^-} f'(x)\)) must match the derivative from the right (\(\lim_{x\rightarrow a^+} f'(x)\)).
• If these derivatives differ, the function is not differentiable at that point.

In the case of \(f(x) = x + |x|\) at \(x = 0\), the left-hand derivative is 0, while the right-hand derivative is 2. Since these do not align, the function is not differentiable at \(x = 0\). Thus, although the function is continuous at \(x = 0\), its lack of a unique tangent there means it is not differentiable.

Functions

Functions are the backbone of calculus, representing relationships between sets of values, typically involving real numbers. A function is defined by an expression like \(f(x) = x + |x|\), where \(f(x)\) denotes the output when \(x\) is input.

Functions exhibit various properties, among which continuity and differentiability are vital.

• A continuous function ensures there are no sudden jumps or breaks in the graph, maintaining a 'smooth' connection of values over an interval.
• A differentiable function, on the other hand, means its graph can be drawn without lifting your pen, exhibiting a smooth curve without sharp edges or corners.

The function \(f(x) = x + |x|\) provides a classic example to illustrate these properties. Despite being smooth in its transition at \(x = 0\) (continuity), its sharp 'corner' at that exact point (lack of differentiability) highlights the nuances of calculus concepts, showing that continuity doesn't always imply differentiability.