The problem is what do you mean by 'is set'?
In vanilla javascript you can do:
element.style.height
This will return an empty string if no height has been set INLINE.
However, if a height has been applied via a stylesheet, it will still return an empty string.
The problem is, if you return a computed height by either .height() in jQuery or window.getComputedStyle(element).height in Javascript, then there is no way of telling if it was calculated by applying a style sheet (what you would call 'having a height property set'), or was generated by extending the height of the element to fit its contents (which you'd call 'not having a height property set').
---------------------Update----------------------
To make it clearer, I'm trying to see if the height of a div is a computed height, or if it a height that was defined in css.
A div can contain the height of it's children, or it can have a height set specifically on the div. The heights for either the children or the div in question can be set in a CSS file, style tag on the page, or on the div itself.
I'm trying to see if the div has a css set height property.
---------------------Update----------------------
I understood your question, but perhaps my answer was a bit opaque.
There is nothing you can call that will tell you if an element has a style property applied to it by a style sheet. In other words, you can't do what you want to do.
The only thing you can find out (via element.style) is if there is an inline style declared.
getComputedStyle will tell you how high an element currently is, but it won't tell you how it got that way.