The function f(x) = x^2 defined on the interval [0, 1] is a quadratic function that takes any real number x within this range and maps it to its square. This specific function is particularly notable because it exhibits continuous behavior and serves as a prime example when discussing uniformly continuous functions, which means that the rate of change of the function does not vary significantly over this closed interval.
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The function f(x) = x^2 is continuous on the closed interval [0, 1], which ensures that it does not have any breaks or jumps in this range.
For any two points x1 and x2 in [0, 1], the absolute difference |f(x1) - f(x2)| can be made small by making |x1 - x2| small, showcasing uniform continuity.
The maximum value of f(x) on [0, 1] occurs at the endpoint x = 1, where f(1) = 1, while the minimum occurs at x = 0 with f(0) = 0.
The derivative of f(x) = x^2 is f'(x) = 2x, which is bounded on [0, 1], further confirming that the function is uniformly continuous.
Using the definition of uniform continuity, for any ε > 0, one can find a δ > 0 such that if |x1 - x2| < δ, then |f(x1) - f(x2)| < ε holds true for all x1, x2 in [0, 1].
Review Questions
How does the behavior of f(x) = x^2 on [0,1] demonstrate the concept of uniform continuity?
The function f(x) = x^2 is uniformly continuous on the interval [0, 1] because it satisfies the condition where for every ε > 0 there exists a δ > 0 such that if |x1 - x2| < δ for any x1 and x2 in [0, 1], then |f(x1) - f(x2)| < ε. This means regardless of where you are in the interval, as long as you keep points sufficiently close together in terms of their input values (x), the outputs (f(x)) will also be close together. The boundedness of its derivative reinforces this property.
What characteristics make f(x) = x^2 a classic example of a uniformly continuous function on a closed interval?
f(x) = x^2 exemplifies uniform continuity due to its smooth and continuous nature over the closed interval [0, 1]. It has no discontinuities or sharp turns, ensuring that small changes in input lead to predictable changes in output. The fact that its derivative is bounded reinforces that no matter how close two inputs are within this interval, their outputs will also remain close, fulfilling the definition of uniform continuity effectively.
Evaluate how understanding the uniform continuity of f(x) = x^2 on [0, 1] could influence broader mathematical concepts or applications.
Recognizing that f(x) = x^2 is uniformly continuous on [0, 1] can have significant implications in various areas such as numerical analysis and approximation theory. This understanding ensures that methods like Riemann sums or numerical integration will yield reliable results since small changes in input values won't produce drastic variations in output. Furthermore, such foundational concepts can aid in solving differential equations and optimization problems where continuity plays a critical role. Thus, mastering these properties lays a strong groundwork for tackling more complex mathematical challenges.
A property of functions where, for any given small positive distance, there exists a corresponding small positive distance in the domain such that the difference in function values is always within the specified distance.
A function is continuous if, intuitively, you can draw its graph without lifting your pencil from the paper, meaning there are no breaks or jumps in the function's values over its domain.
Closed Interval: An interval that includes its endpoints, denoted as [a, b], where both a and b are part of the interval.