Question

(a) Let \(f_{1}\) and \(f_{2}\) be continuous at \(x_{0}\) and define $$ F(x)=\max \left(f_{1}(x), f_{2}(x)\right) $$ Show that \(F\) is continuous at \(x_{0}\) (b) Let \(f_{1}, f_{2}, \ldots, f_{n}\) be continuous at \(x_{0}\) and define $$ F(x)=\max \left(f_{1}(x), f_{2}(x), \ldots, f_{n}(x)\right) $$ Show that \(F\) is continuous at \(x_{0}\).

Short answer

Answer: Yes, the maximum of a finite number of continuous functions is also continuous.

Step-by-step solution

Solution for (a)

The maximum of two continuous functions is also continuous. If we pick any: |F(x) − F(x_0)| < ε, we will be able to find a δ > 0 such that |x − x_0| < δ ⇒ |f_i(x) − f_i(x_0)| < ε for both i = 1, 2. Since |f_i(x) − f_i(x_0)| are less than ε, we are guaranteed that |max {f_1(x), f_2(x)} − max{f_1(x_0), f_2(x_0)}| < ε. Thus F(x) is continuous at x_0.

Solution for (b)

The proof for when there are 'n' functions is essentially the same as the proof for two functions, except that max{f_1(x), f_2(x), ..., f_n(x)} is the maximum of the n functions rather than just two functions. Because each of the functions f_i (i = 1, 2, ..., n) is continuous at x_0, we can pick any |F(x) − F(x_0)| < ε and find a δ > 0 such that |x − x_0| < δ will cause |f_i(x) − f_i(x_0)| < ε for each i. This will ensure that |max {f_1(x), f_2(x), ..., f_n(x)} − max{f_1(x_0), f_2(x_0), ..., f_n(x_0)}| < ε; hence F(x) is continuous at x_0. The general principle here is that the maximum of a finite number of continuous functions is also continuous.

Key concepts

Maximum of Functions

When dealing with the maximum of functions, especially in the context of continuity, it is crucial to understand how the maximum function operates with multiple inputs. The maximum function takes a set of values and returns the largest among them. For example, given two functions, \( f_1(x) \) and \( f_2(x) \), the expression \( F(x) = \max(f_1(x), f_2(x)) \) renders the greater value of these two at each point \( x \).
This means, at every point \( x \), you are evaluating both \( f_1(x) \) and \( f_2(x) \) and simply choosing the larger output. Therefore, if both functions are continuous individually, their maximum will also follow a path that reflects slight variations, ensuring continuity.
In scenarios where more functions are involved, this definition holds. So for functions \( f_1, f_2, \, \ldots, f_n \), the maximum function \( F(x) = \max(f_1(x), f_2(x), \, \ldots, f_n(x)) \) again selects the largest value among these at each \( x \). This very nature of selection makes analyzing maximum functions especially applicable in real-world scenarios where it’s often necessary to choose the best possible outcome.

Real Analysis

Real Analysis is a field of mathematics that deals with real numbers and real-valued sequences and functions. It provides a rigorous foundation for calculus and focuses on understanding the behavior of real-valued functions in great detail. Within this field, the continuity of functions is a fundamental concept.
Through real analysis, the properties of continuity can be explored through limits, sequences, and series. It delves deeply into understanding how small changes in input can affect the function's output and how these behaviors can be generalized for multiple functions. By studying real analysis, students and researchers can discern how certain operations, like taking the maximum of functions, can influence continuity and the understanding thereof.
Moreover, real analysis not only deals with well-behaved functions that are continuous everywhere but also explores more complex cases where continuity might present challenges. By understanding these nuances, one gains insight into how functions converge, behave, and interact on a deeper level.

Continuous Functions

Continuous functions are a central topic in calculus and real analysis, marking the functions that maintain consistent output without abrupt changes as inputs change gradually. A function \( f \) is described as continuous at a point \( x_0 \) if, for every small \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x \) satisfying \( |x - x_0| < \delta \), we have \( |f(x) - f(x_0)| < \epsilon \).
This definition can be interpreted intuitively: small changes in \( x \) result in small changes in \( f(x) \). When evaluating continuous functions, they do not exhibit sudden jumps or breaks and are paramount in many areas of mathematics due to their predictable behavior.
The concept extends beyond single functions to operations involving multiple functions. For instance, when considering the maximum of multiple continuous functions, as in the exercise above, the resultant function (the maximum function) itself is continuous. This property is crucial when dealing with real mathematical models, such as determining the maximum temperature from multiple sensors, ensuring that our calculations and predictions remain reliable and accurate.