# Thread: What is the definition of Single-Peaked function?

1. ## What is the definition of Single-Peaked function?

Dear all,

I have some confusion on the definition of Single-Peaked function. Can anybody tell me the official definition of it?

It is intuitive to understand "Single-Peaked". However, what if the function has a flat part with its level lower than the peak? Can we say this function still Single-Peaked?

In other words, do we require the strict monotonicity on both sides of the peak for it to be "Single-Peaked"?

2. Originally Posted by analysismath
"Single-Peaked"?

I've never heard this before...

3. My guess would be that a single peaked function would be one of which has only 1 maximum value. An example of this would be an inverted quadratic, a cubic with no points of inflection or even an inverted quartic with a point of inflection. This could extend further when looking at the function that generates the bell curve for a normal distribution.

Given the function has only one global maximum, if it has a flat part somewhere else other than the only "peak" point, would we still count it as the "Single-Peaked"?

Equivalently, do we require strict monotonicity on both sides of the unique global maximum?

5. Originally Posted by analysismath

Equivalently, do we require strict monotonicity on both sides of the unique global maximum?
Are we concerned with the entire number line?

6. Originally Posted by analysismath

Given the function has only one global maximum, if it has a flat part somewhere else other than the only "peak" point, would we still count it as the "Single-Peaked"?
I would say yes given the definition of a maximum is a point that has a positive gradient preceeding it and a negative gradient afterwards if the flat part did not adhere to these conditions itself.