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Revisiting Fixed Point: What does it mean in the world of computer science?

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Problem Detail: 

A while ago I asked Fixed point, what does it mean in the world of computer science?

While the answers did help me to understand what Fixed point meant, the answers left me in a murky world when ever I thought about what a fixed point meant. In other words, no clear definition poped into my mind and I would wonder about in thought trying to match up a definition with what I was trying to learn or understand.

Today I was watching the video Categories for the Working Programmer by Jeremy Gibbons in which he stated: (Time index: 33:00)

It's fix point because it's the solution of a recursive equation.

And I thought that makes sense; is it really that simple?

So the question is:

Can the definition of fix point be thought of simply as

"It's the solution of a recursive equation."

Asked By : Guy Coder
Answered By : Yuval Filmus

A fixed point of a function $f$ is an element $x$ satisfying $f(x) = x$. A solution to a recursive equation can be put in that form.

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