# What is the definition of a Critical Point?

I was reading a book on Calculus, by Michael Spivak. There they mention that points where the derivative is equal to zero are called critical points. They nowhere mention that where the derivative does not exist will be a critical point. When I Googled it, they say critical points are those where the function derivative is either zero or the derivative does not exist. Which definition should I take?

• Different people use the term in somewhat different ways. – André Nicolas Jul 28 '15 at 19:34
• my question is whom to believe, apostol calculus ... there also it is not given that where the derivative is not defined, those are also critical points. I was making short notes and I want to follow a book where they give all the definition which are correct and universally accepted. – Abhishek Jul 28 '15 at 19:37
• As intimated above, "critical point" is not one of those "universally accepted" terms. – David Mitra Jul 28 '15 at 19:41
• as fas as an exam is concerned... what should i tell my students, when I am not very sure about the same. – Abhishek Jul 28 '15 at 19:43
• "Might" is too vague to use in definitions. – Robert Israel Jul 28 '15 at 20:38

When the derivative is 0 at a point $(x,y)$, that point is critical. When a derivative does not exist, there might be no single point that can be labeled as critical. For example, the function $x, x\in (-\infty, 0)$ and $x+3, x\in [0, \infty)$. The derivative does not exist at $x=0$, however there is no single point that can be labeled as critical. So in my opinion, if $\lim_{x\to a} f(x)$ exists, then that is a critical point.