# Graphing

If you want to have a complete graph of a function f, then these are all of the things that you should make sure show up:
• x = 0, if f is defined at that point;
• x → −∞, if f is defined in that direction;
• x → ∞, if f is defined in that direction;
• x → c, if f is defined in that direction, whenever f is undefined or discontinuous at c;
• x → c+, if f is defined in that direction, whenever f is undefined or discontinuous at c;
• x = c, if f is defined at that point, whenever f is undefined approaching c from either direction (or both);
• x = c, whenever f(c) = 0;
• x = c, whenever f′ is undefined or discontinuous at c, if f is defined there;
• x = c, whenever f′(c) = 0;
• x = c, whenever f′ is undefined or discontinuous at c, if f is defined there;
• x = c, whenever f′(c) = 0;
This should be sufficient whenever f is a twice-differentiable function whose domain is an interval, or more generally whenever f is piecewise twice-differentiable: a piecewise-defined function in which the domain of each piece is an interval and in which each piece is twice-differentiable except possibly at its endpoints. (There are weirder functions that can't be put in this form, but you shouldn't have to deal with them in this class.)
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