Question

Let \(f, g: \mathbb{R}^{n} \rightarrow \mathbb{R}\) be continuous at a point \(a \in \mathbb{R}^{n} .\) Define \(h, k: \mathbb{R}^{n} \rightarrow \mathbb{R}\) by $$ h(x)=\sup \\{f(x), g(x)\\} \quad \text { and } \quad k(x)=\inf \\{f(x), g(x)\\}, x \in \mathbb{R}^{n} $$ Show that \(h\) and \(k\) are continuous at \(a\).

Short answer

Short Answer: To prove the continuity of the supremum function \(h\) and the infimum function \(k\) at a point \(a\), we used the sequential definition of continuity. We have shown that for any sequence converging to \(a\), the limit of \(h(x_n)\) and \(k(x_n)\) is equal to \(h(a)\) and \(k(a)\), respectively. This establishes that both \(h\) and \(k\) are continuous at the point \(a\).

Step-by-step solution

Define the Sequential Continuity

A function \(f\) is continuous at a point \(a\) if, for every sequence \((x_n)\) converging to \(a\), the following holds: $$ \lim_{n \rightarrow \infty} f(x_n) = f(a) $$ We will use this definition to prove the continuity of \(h\) and \(k\) at \(a\).

Prove Continuity of \(h\)

We have \(h(x) = \sup\{f(x), g(x)\}\). Let \((x_n)\) be any sequence converging to \(a\). Since \(f\) and \(g\) are continuous at \(a\), we have: $$ \lim_{n \rightarrow \infty} f(x_n) = f(a) \quad \text{and} \quad \lim_{n \rightarrow \infty} g(x_n) = g(a) $$ Now, let's consider the sequence \(h(x_n) = \sup\{f(x_n), g(x_n)\}\). We will show that \(\lim_{n\rightarrow\infty} h(x_n) = h(a)\). By definition, for each \(x_n\) either \(f(x_n) \ge g(x_n)\) or \(g(x_n) \ge f(x_n)\). Therefore, there exist subsequences \((x_{n_1})\) and \((x_{n_2})\) of \((x_n)\) such that \(f(x_{n_1}) \ge g(x_{n_1})\) and \(g(x_{n_2}) \ge f(x_{n_2})\). Note that \((x_{n_1})\) and \((x_{n_2})\) also converge to \(a\). As a result, we have: $$ \lim_{n_1\rightarrow\infty} f(x_{n_1}) = \lim_{n_2\rightarrow\infty} g(x_{n_2}) = f(a) \quad \text{and} \quad g(a) $$ Therefore, we have $$ \lim_{n\rightarrow\infty} h(x_n) = \sup\{f(a), g(a)\} = h(a) $$ This establishes the continuity of \(h\) at \(a\).

Prove Continuity of \(k\)

We have \(k(x) = \inf\{f(x), g(x)\}\). The proof of continuity for \(k\) follows a similar procedure as for \(h\). Let \((x_n)\) be any sequence converging to \(a\). Since \(f\) and \(g\) are continuous at \(a\), we have: $$ \lim_{n \rightarrow \infty} f(x_n) = f(a) \quad \text{and} \quad \lim_{n \rightarrow \infty} g(x_n) = g(a) $$ Now, let's consider the sequence \(k(x_n) = \inf\{f(x_n), g(x_n)\}\). We will show that \(\lim_{n\rightarrow\infty} k(x_n) = k(a)\). By definition, for each \(x_n\) either \(f(x_n) \le g(x_n)\) or \(g(x_n) \le f(x_n)\). Therefore, there exist subsequences \((x_{n_1})\) and \((x_{n_2})\) of \((x_n)\) such that \(f(x_{n_1}) \le g(x_{n_1})\) and \(g(x_{n_2}) \le f(x_{n_2})\). Note that \((x_{n_1})\) and \((x_{n_2})\) also converge to \(a\). As a result, we have: $$ \lim_{n_1\rightarrow\infty} f(x_{n_1}) = \lim_{n_2\rightarrow\infty} g(x_{n_2}) = f(a) \quad \text{and} \quad g(a) $$ Therefore, we have $$ \lim_{n\rightarrow\infty} k(x_n) = \inf\{f(a), g(a)\} = k(a) $$ This establishes the continuity of \(k\) at \(a\). In conclusion, both \(h\) and \(k\) are continuous at the point \(a\).

Key concepts

Supremum and Infimum Functions

The concepts of supremum and infimum are essential in mathematical analysis when dealing with sets and functions. The supremum, or "least upper bound," is the smallest number that is greater than or equal to every element of a set. You can think of it as the ceiling that the values cannot exceed. Conversely, the infimum, or "greatest lower bound," is the largest number that is less than or equal to every element of a set - like the floor that the values cannot fall below.

For functions, considering the supremum or infimum involves evaluating values over a domain. For instance, if you have two continuous functions, such as in our exercise with functions \(f\) and \(g\), combining them using \(\sup\) and \(\inf\) provides new functions. In this scenario, we associate with each point \(x\) the larger of the two function values \(f(x)\) or \(g(x)\) to define \(h(x)=\sup\{f(x), g(x)\}\), and \(k(x)=\inf\{f(x), g(x)\}\) uses the smaller value.

Understanding these concepts is vital, as it enables one to understand how limits and boundaries apply to more complex function combinations, crucial for solving continuity and boundedness problems.

Sequential Continuity

Sequential continuity is a more intuitive way of understanding the continuity of functions via sequences. In simple terms, a function \(f\) is continuous at a point \(a\) if whenever you take any sequence \((x_n)\) that converges to \(a\), the values of \(f(x_n)\) must converge to \(f(a)\). This reflects the idea that there are no jumps or breaks in the function at that point.

In the case of our exercise, both functions \(f\) and \(g\) are continuous at the point \(a\). This implies that for any sequence \((x_n)\) getting closer and closer to \(a\), the sequences \((f(x_n))\) and \((g(x_n))\) will converge to \(f(a)\) and \(g(a)\) respectively. Thus, by examining subsequences formed by the supremum (maximum between the two) and infimum (minimum), we establish that both \(h(x)\) and \(k(x)\) are continuous.

Sequential continuity provides a constructive proof technique by leveraging sequences, making it easier to show continuity using well-understood limits.

Proof Techniques in Analysis

Proving the continuity of functions like \(h\) and \(k\) involves rigorous and logical reasoning. Key proof techniques include direct proofs, contraposição, and using properties of limits and continuity. In the exercise at hand, the continuity of the functions \(h\) and \(k\) at point \(a\) is proven by harnessing the concept of sequential continuity.

Here is a look at how we structured the proof for \(h\). First, identify the sequences from \(h(x_n) = \sup\{f(x_n), g(x_n)\}\). By realizing every element converges due to the continuity of \(f\) and \(g\), it naturally follows that the sequences of the suprema will converge to \(\sup\{f(a),g(a)\}\).

The methodical application of subsequences \((x_{n_1})\) and \((x_{n_2})\) allows us to isolate where each function \(f(x)\) exceeds \(g(x)\) or vice versa, thus simplifying convergence implications. This detailed work safeguards the assertion of continuity.

For \(k\), the approach is analogous. We reverse the inequality conditions and validate for each converging subsequence via the infimum, illustrating the robust overlap in techniques across different scenarios.

• Define your sequences and limits clearly.
• Break down each step logically, relying on previous function properties.

Such techniques are foundational to advanced analysis, showcasing the elegance of mathematical proofs.