# (x)=max{f(x),g(x)} continuous functions

• QUBStudent
In summary, the equation in the title states that if f and g are continuous functions, then M(x) is also continuous. To prove this, we can use the fact that max{a, b} = (a + b + |a - b|)/2. This transforms the problem into showing that (f(x) + g(x) + |f(x) - g(x)|)/2 is continuous.
QUBStudent
lo,
I've got a quick q about the equation in the title, I've been asked to show/prove by analysis, that if f and g are continuous functions then M(x) is also continuous, it seems pretty intuitive but i just don't know how they want us to prove it, any help would be gr8

Use the fact that max{a, b} = (a + b + |a - b|)/2.

how can i use this fact?

It transforms the problem of showing that max{f(x), g(x)} is continuous into showing that (f(x) + g(x) + |f(x) - g(x)|)/2 is continuous.

thx, think i get it now

## 1. What is the definition of a continuous function?

A continuous function is a type of mathematical function that is defined for all points on its domain and has a consistent graph with no breaks or gaps. This means that the function can be drawn without lifting the pencil from the paper.

## 2. How is the maximum of two continuous functions calculated?

The maximum of two continuous functions, (x)=max{f(x),g(x)}, is calculated by comparing the values of the two functions at each point on their respective domains, and choosing the larger value as the output for that point.

## 3. Can a continuous function have more than one maximum value?

Yes, a continuous function can have multiple maximum values. This can happen when there are multiple points on the graph where the function reaches its peak value.

## 4. What is the significance of the maximum of two continuous functions in real-life applications?

The maximum of two continuous functions is often used in decision making processes, where the maximum value represents the best or optimal choice. It can also be useful in determining the highest possible value for a given situation.

## 5. How can the continuity of a function impact its maximum value?

The continuity of a function can greatly affect its maximum value. A continuous function with no breaks or gaps will have a well-defined maximum value, while a discontinuous function may have multiple maximum values or no maximum value at all.

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