1. ## Re: Open Interval differention

Originally Posted by rgjf1307
The format Plato used earlier is along the lines of what I was hoping to achieve, I was just wondering how, using the same notation, would you define a critical point of a differentiable function?
A function has a critical point if its derivative fails to exist at the point or if its derivative is zero at that point.

If a function $f$ is not continuous at $x_0$ then it has no derivative at $x_0$ so that is a critical value.

On the other hand, the slope on the left at $x_0$ may not be the slope on the right at $x_0$, in that case we call that a "spike". There is no derivative there.

2. ## Re: Open Interval differention

You say you're taking a calculus course and you refer to a text, but haven't seen a definition of "derivative"? Before you report your instructor, you should make absolutely sure that you're right. I'm not saying you're lying, but rather that you missed the definition in the book. Even though that seems unlikely, a calculus book not having a definition of derivative seems even less likely.

Wikipedia is also helpful for basic definitions like this.

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