Proving Convexity of a Function on an Open Interval

In summary, to prove that a convex function on an open interval I, which is a subset of R, can be differentiated on the whole interval, it must be shown that the function is differentiable from the left and from the right. This means that the derivatives from both directions do not have to be equal, as illustrated by the example of f(x)=|x| on I=(-1,1).
  • #1
Jonas Rist
7
0
Hello,

how can I proof this:
given: a convex function on an open interval I,which is a subset of R.
I have to show that the function can be differentiated on the whole interval.
I already proved the following for a<b<c, a,b,c in I:
f(b)-f(a))/(b-a)<(f(c)-f(a))/(c-a)<(f(c)-f(b))/(c-b).

Thanks for help!
Jonas
 
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  • #2
Isn't f(x)= |x| convex on I= (-1,1)?
 
  • #3
Ah, sorry,
actually one has to show that the function is differentiable from the left and from the right(but these derivatives don´t have to be equal, as your example shows clearly).
Jonas
 

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

A function is convex if, for any two points on its graph, the line segment connecting them lies above or on the graph. In other words, the function is always "curving upwards" and does not have any "dips".

2. What is an open interval?

An open interval is a set of real numbers between two given values, where the endpoints are not included in the interval. For example, the open interval (0,1) would include all numbers between 0 and 1, but not 0 or 1 themselves.

3. How can you prove the convexity of a function on an open interval?

To prove the convexity of a function on an open interval, you can use the definition of convexity and show that the line segment connecting any two points on the graph lies above or on the graph. This can be done by using calculus techniques, such as finding the second derivative of the function and showing that it is always positive on the given interval.

4. What is the importance of proving convexity of a function?

Proving convexity of a function is important because it guarantees certain properties of the function, such as uniqueness of the global minimum. It also allows us to use convex optimization techniques to find the optimal solution to a problem.

5. Can a function be convex on a closed interval but not on an open interval?

Yes, a function can be convex on a closed interval but not on an open interval. This is because the definition of convexity for a closed interval includes the endpoints, while the definition for an open interval does not. Therefore, the function may have a "dip" at one of the endpoints, making it not convex on the open interval but still convex on the closed interval.

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