# The graph of a continuous function has zero content

• Castilla
In summary, the proof states that if f is uniformly continuous in [a, b], then the graph of the function, v. gr. the set { (x, f(x)) / a < x < b }, has zero content.
Castilla
Good morning, I am trying to understand why this statement is true:

"If f, a real function of real variable, is continuous on the closed interval [a, b], then the graph of the function, v. gr. the set { (x, f(x)) / a < x < b }, has zero content".

I have found this proof in the web:

As f is uniformly continuous in [a, b], then for every natural n there is a natural m such that if x, y belong to [a, b] and |x - y| < (b - a)/m, then
|f(x) - f(y)| < 1/n.

Then the next rectangles in RxR (j= 1,2,...,m) cover the set { (x, f(x)) / a < x < b }:

[ a + (1/m)(j-1)(b-a) - 1/mn, a + (1/m)j(b-a) + 1/mn] x

[ f(a + (1/m)(j-1)(b-a)) - 1/n - 1/mn, f(a + (1/m)(j-1)(b-a)) + 1/n + 1/mn]

Is simple to proof that this collection of rectangles has zero content, because its area -> 0 when n -> +oo.

But I do not understand how can we be sure that said rectangles cover the graphic of the function f.

Let's take some x, y of [a,b] such that |x-y| < (b-a)/m. Then there is some j such that x, y belong to

[ a + (1/m)(j-1)(b-a) - 1/mn, a + (1/m)j(b-a) + 1/mn]

But I do not see how this would asure that the values f(x) and f(y) belong to the real interval

[ f(a + (1/m)(j-1)(b-a)) - 1/n - 1/mn, f(a + (1/m)(j-1)(b-a)) + 1/n + 1/mn]

I know that the values f(x) and f(y) fulfill |f(x) - f(y)| < 1/n, but this does not implies that f(x) and f(y) belong to the real interval

[ f(a + (1/m)(j-1)(b-a)) - 1/n - 1/mn, f(a + (1/m)(j-1)(b-a)) + 1/n + 1/mn]

So how, please, how can we asure that those collection of rectangles in RxR covers the graph of the function f ?

Just draw a picture.

Matt:

Supose I take x1 and x2 in the x-axis such that both belong to [a,b] and |x1 - x2| < (b-a)/m.

There is necesarily a "j" such that x1, x2 belong to the real interval [a + {(1/m)(j-1)(b-a)} - (1/mn), a + {(1/m)j(b-a)} + (1/mn)].

But this does not imply that f(x), f(y) belong to the real interval

[f(a + (1/m)(j-1)(b-a)) - (1/n) - (1/mn), f(a + (1/m)(j-1)(b-a)) + (1/n) + (1/mn)]

Maybe both are < f(a + (1/m)(j-1)(b-a)) - (1/n) - (1/mn)
or maybe > f(a + (1/m)(j-1)(b-a)) + (1/n) + (1/mn).
how can I know?? The drawing doesn't makes this imposible.

Er, no I don't believe that you can necessarily state x1 and x2 lie in the same interval. And that isn't important.

Split the region between a and b into m smaller intervals. The image of some part each interval lies in some region of some given length using the uniform continuity of f in a nice wa, so it all covers (like i said draw a picture for a simple example to get the idea, try f(x)=x on [0,1]), so the area we can work out in terms of m and n. Then we can probably do soemthing like let m tend to infinity, then n tend to infinity and we are done.

Last edited:

## 1. What is a continuous function?

A continuous function is a mathematical function that has a continuous graph, meaning that there are no gaps or jumps in the graph. This means that the function can be drawn without lifting the pen from the paper.

## 2. What does it mean for a function to have zero content on its graph?

If a function has zero content on its graph, it means that the area under the curve is equal to zero. This could happen when the function intersects the x-axis at a point where the y-value is also zero, or when the function has a constant value of zero.

## 3. Why is it important for a continuous function to have zero content on its graph?

Having zero content on its graph is important for a continuous function because it indicates that the function is not crossing the x-axis and has a constant value of zero. This can be useful in certain applications, such as finding the roots of a function.

## 4. Can a continuous function have zero content at multiple points on its graph?

Yes, a continuous function can have zero content at multiple points on its graph. This could happen if the function has multiple x-intercepts where the y-value is also zero, or if the function has a constant value of zero for a certain interval.

## 5. What are some real-life examples of continuous functions with zero content on their graphs?

Real-life examples of continuous functions with zero content on their graphs include a constant temperature over a certain period of time, a stationary object's position over time, and a sound wave with constant amplitude. In these cases, the graph of the function would have a horizontal line at the x-axis, indicating zero content.

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