# What does a real-valued function on [0,1] mean ?

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In summary, a real-valued function on [0,1] is a mathematical function that maps values from the interval [0,1] to real numbers. It is represented as f(x) and only takes and outputs real numbers. The interval [0,1] is commonly used for real-valued functions and they can be represented graphically as a curve on a coordinate plane. Some examples of real-valued functions on [0,1] include linear, quadratic, trigonometric, and exponential functions.
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## Homework Statement

As the tittles says "What does a real-valued function on [0,1] mean ?"

## The Attempt at a Solution

Does it mean $$0 \leq f(x) \leq 1$$ for all x

Or does it mean$$x \in[0,1]$$ and f(x) need not be bounded ?

Or does it mean something else ?

I am trying to write a proof which talks about a real valued function on [0,1] and I need to understand to eventually write the proof.

It just means for x in [0,1] f(x) is a real number. It doesn't imply bounded or anything else.

Alright, thanks.

## What is a real-valued function on [0,1]?

A real-valued function on [0,1] is a mathematical function that takes values from the interval [0,1] and maps them to real numbers. It can be represented as f(x), where x is an input from the interval [0,1] and f(x) is the corresponding output.

## What does it mean for a function to be real-valued?

A real-valued function is one that only takes real numbers as inputs and outputs real numbers. This means that the graph of the function will only contain points on the real number line.

## Why is the interval [0,1] often used for real-valued functions?

The interval [0,1] is often used for real-valued functions because it is a commonly used range of values in mathematics and it allows for easy comparison and analysis of different functions.

## How is a real-valued function represented graphically?

A real-valued function can be represented graphically as a curve on a coordinate plane. The x-axis represents the input values and the y-axis represents the corresponding output values. The points on the curve show the relationship between the inputs and outputs of the function.

## What are some examples of real-valued functions on [0,1]?

Some examples of real-valued functions on [0,1] include linear functions, quadratic functions, trigonometric functions, and exponential functions. These functions can be represented as f(x) = mx + b, f(x) = ax^2 + bx + c, f(x) = sin(x), and f(x) = e^x, respectively.

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