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Magazine Chapter 1: Problem 39
Prove that \(f(x)=x^{3 / 5}\) is continuous everywhere, carefully justifying each step.
Short Answer
Expert verified
The function \(f(x) = x^{3/5}\) is continuous everywhere on \(\mathbb{R}\).
Step by step solution
01
Recall the Definition of Continuity
A function \(f(x)\) is continuous at a point \(a\) if the following three conditions are satisfied: \(f(a)\) is defined, the limit \(\lim_{{x \to a}} f(x)\) exists, and \(\lim_{{x \to a}} f(x) = f(a)\). To prove continuity everywhere, these conditions must hold for all \( x \in \mathbb{R} \).
02
Analyze the Function's Expression
The function is \(f(x) = x^{3/5}\), which is a power function with an exponent of \(\frac{3}{5}\) where \(p = 3\) and \(q = 5\) are both integers. Observe that power functions \(x^{n/m}\) with \(m\) odd are continuous on \(\mathbb{R}\). Thus, we need to verify especially for the problematic point, which is at \(x = 0\).
03
Check Continuity at Zero
Evaluate \(f(0)\): \(f(0) = 0^{3/5} = 0\). Next, calculate \(\lim_{{x \to 0}} x^{3/5}\). Since \(x^{3/5}\) is defined for all \(x\), the expression approaches zero as \(x\) approaches zero from both sides. Thus, \(\lim_{{x \to 0}} x^{3/5} = 0\). Since both the function value and the limit agree at zero, we conclude \(f\) is continuous at \(x = 0\).
04
Confirm Continuity Elsewhere
For any \(x eq 0\), since \(f(x) = x^{3/5}\) is composed of continuous operations (taking a root and raising to a power), \(f(x)\) remains continuous. This includes evaluating \(\lim_{{x \to a}} x^{3/5}\) for any arbitrary \(aeq 0\), which will similarly show \(\lim_{{x \to a}} f(x) = f(a)\) under the rules of continuity properties of the components of the function.
05
Conclude that the Function is Continuous Everywhere
Since all points, including \(x = 0\) and \(x eq 0\), satisfy the conditions of continuity and no piece of the function breaks these conditions, \(f(x) = x^{3/5}\) is indeed continuous over all real numbers \(\mathbb{R}\). Therefore, the initial claim is proven true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit of a Function
The concept of a limit is essential in understanding continuity in calculus. When we talk about the limit of a function \( f(x) \) as \( x \) approaches a point \( a \), we are interested in the value that \( f(x) \) gets closer and closer to as \( x \) nears \( a \). This limit is denoted as \( \lim_{{x \to a}} f(x) \).
Here are a few simple points to consider when evaluating limits:
Here are a few simple points to consider when evaluating limits:
- The limit could be a specific number at which the function stabilizes as \( x \) approaches \( a \).
- If a function has a limit at a point, it does not necessarily mean the function is defined at that point; there might be a hole or a jump.
- Evaluating limits often involves simplifying the function or using approximation methods for complex expressions.
Power Functions
Power functions are a major category of functions defined by expressions of the form \( f(x) = x^{n/m} \), where \( n \) and \( m \) are integers. These functions have unique characteristics based on the values of \( n \) and \( m \).
Understanding power functions is vital for students learning calculus:
Understanding power functions is vital for students learning calculus:
- If \( m \) (the denominator) is odd, the function is continuous across all real numbers, meaning it will not break or have gaps.
- If \( m \) is even, special care must be taken as you can't directly evaluate negative bases for real roots (such as square roots).
- The shape and direction of the graph are determined largely by the exponents \( n/m \).
Continuity at a Point
The concept of continuity at a point involves ensuring a function behaves predictably within a small interval around that point. Specifically, a function \( f(x) \) is termed continuous at a point \( a \) if it satisfies three conditions:
- \( f(a) \) is defined 鈥� the function must have a real output at \( a \).
- The limit \( \lim_{{x \to a}} f(x) \) must exist 鈥� the function鈥檚 values should stabilize as they approach \( a \).
- The limit must equal the function value at that point: \( \lim_{{x \to a}} f(x) = f(a) \).
- \( f(0) \) is defined and equals 0.
- The limit \( \lim_{{x \to 0}} x^{3/5} \) exists and is 0.
- Both the function value and the limit match at 0, fulfilling all conditions of continuity.
