What determines the differentiability of a function? The premise for my confusion is that my textbook states that a point is differentiable if the limit at that point exists and it is continuous. However, a cusp is continuous, the limit exists, and yet, is not differentiable...
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The statement that "the limit exists" doesn't refer to the limit of the function itself, $$ \lim_{x \to x_0} f(x), $$ but instead to the limit $$ \lim_{h \to 0} \frac{f(x_0 + h) - f (x_0)}{h}. $$ In particular, for a function with a cusp, this limit will differ as $h \to 0$ from above and from below, and so the limit does not exist.
For example, for $f = |x|$ and $x_0 = 0$, we have $$ \lim_{h \to 0_+} \frac{|h| - 0}{h} = \lim_{h \to 0_+} \frac{h}{h} = 1, $$ but $$ \lim_{h \to 0_-} \frac{|h| - 0}{h} = \lim_{h \to 0_+} \frac{-h}{h} = -1. $$
If you consider both points of a cusp to be two separate functions and differentiate those two functions, and concatenate those results, you will find that to not be continuous.
In other words, the slopes of the two functions as they approach the cusp are not the same. To be differentiable, all points on a function must have the same slope when approached in both directions.