# How to Prove Continuity of max{f(x), g(x)} at a Point c

• benya7thmix
In summary, the conversation is discussing a problem from a sample test in an "Introduction to Real Analysis" class. The problem involves proving the continuity of a function h(x) using two cases and utilizing known continuity properties for expressions of functions. The solution involves showing that h(x) can be represented as the average of two functions plus the absolute value of their difference, and then applying known continuity properties.
benya7thmix
this is a problem from a sample test in my "Introduction to Real Analysis" class. i don't know how to even start thinking about this one. any hints?

Let f, g : R ! R be continuous at c, and let h(x) = max{f(x), g(x)}.

(a) Show that h(x) = (1/2) (f(x) + g(x)) + (1/2) |f(x) − g(x)| for all x in R.
(b) Show that h(x) is continuous at c.

thanks to all posters

For a) Check the two cases:
If f(x) is greater than g(x) at some point x, then h(x) should reduce to f(x).
And likewase if g(x) is greater than f(x) at the point x.

For b), what continuity properties for expressions of functions are you aware of thatmight help you?

## 1. How do you determine the hypothesis for a proof?

The hypothesis for a proof can be determined by carefully analyzing the given statement or problem and identifying the key elements that need to be proven.

## 2. What is the role of counterexamples in a proof?

Counterexamples play a crucial role in a proof as they help to disprove a statement or conjecture. By providing a specific example that goes against the given statement, a counterexample can help to strengthen the validity of a proof.

## 3. How do you choose the appropriate proof technique for a problem?

Choosing the appropriate proof technique depends on the type of problem and the given information. Some common techniques include direct proof, proof by contradiction, and mathematical induction.

## 4. What is the importance of clearly stating the logical steps in a proof?

Clearly stating the logical steps in a proof is important as it helps to make the proof more organized, logical, and easy to follow. This also allows for easier identification of any errors or mistakes in the proof.

## 5. How do you handle a proof that seems unsolvable?

If a proof seems unsolvable, it is important to break down the problem into smaller, more manageable parts. Often, approaching the problem from a different angle or using a different proof technique can also help to solve a seemingly unsolvable proof.

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