# If f:[0,1] -> R is a continuous function, describe f.

• irresistible
In summary, the given problem states that there exists a continuous function f on the interval [0,1], for which the area under its graph from 0 to x is equal to the area under its graph from x to 1 for all values of x in [0,1]. This implies that f must cross the x-axis at some point between 0 and 1, or be the 0 function. Additionally, the fundamental theorem of calculus can be used to show that there exists a function F on [0,1] that satisfies the given condition.
irresistible

## Homework Statement

f:[0,1] $$\rightarrow$$ R is a continuous function such that
$$\int$$f(t)dt (from 0 to x) = $$\int$$ f(t)dt( from x to 1) for all x$$\in$$[0,1] .
Describe f.

## Homework Equations

integral represents area

## The Attempt at a Solution

what ever the function is, I know that the area under the graph from 0 to x is equal to the area under the graph from x to 1.
But how else can I describe f?

One possibility is that f(t) = 0 for all t in [0, 1]. If there are other functions, none come to mind.

Note that the integral is a signed area, so areas under the x-axis are negatively signed. If you draw a picture of a function that is entirely on one side of the axis (positive or negative) and start integrating from 0 to small x, it implies the remaining curve must take a nose-dive in order to account for the small amount of area between 0 and x. On the other hand, if we place x close to 1, it implies the small region between x and 1 must be higher than the previous region in order to account for the larger area, which doesn't agree with the previous scenario (or the function is the 0 function).
This implies that the function must cross the x-axis between 0 and 1 (or be the 0 function). Can you continue from there?
Another less geometric and more algebraic approach is to use the fundamental theorem of calculus to note that there exists some function F on [0,1] such that the first integral is F(x) - F(0) and the second integral is F(1) - F(x), which leads to the same conclusion.

I think so,Thank you

## 1. What does it mean for a function to be continuous?

A continuous function is one in which the output values change smoothly as the input values change. This means that for any small change in the input, there will only be a small change in the output. In other words, there are no sudden jumps or gaps in the graph of a continuous function.

## 2. What is the domain and range of a continuous function?

The domain of a continuous function is the set of all possible input values, in this case [0,1]. The range is the set of all possible output values, which can be any real number (R).

## 3. How is continuity of a function determined?

A function is considered continuous if the limit of the function as x approaches a certain value is equal to the function evaluated at that value. In other words, the function has no breaks or holes in its graph.

## 4. What are some examples of continuous functions?

Examples of continuous functions include polynomials, trigonometric functions, and exponential functions. For instance, f(x) = x^2 is a continuous function on the interval [0,1] because it is a polynomial function.

## 5. How does continuity of a function relate to its differentiability?

If a function is continuous, it does not necessarily mean that it is differentiable. A function is differentiable if it has a well-defined derivative at each point in its domain. However, if a function is differentiable, it must also be continuous.

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