Mathematics

The key is to check differentiability at the boundary where the formula changes—here, $x = 0$—by computing the left and right derivatives and seeing if they match. A corner occurs when both one-sided derivatives exist but differ; a cusp when they exist with opposite signs and infinite magnitude; a vertical tangent when they're both infinite. You'll need to carefully evaluate $\lim_{h \to 0^-} \frac{f(h) - f(0)}{h}$ and $\lim_{h \to 0^+} \frac{f(h) - f(0)}{h}$ to determine which case applies.

The key is to check differentiability at the boundary where the formula changes—here, $x = 0$—by computing the left and right derivatives and seeing if they match. A corner occurs when both one-sided derivatives exist but differ; a cusp when they exist with opposite signs and infinite magnitude; a vertical tangent when they're both infinite. You'll need to carefully evaluate $\lim_{h \to 0^-} \frac{f(h) - f(0)}{h}$ and $\lim_{h \to 0^+} \frac{f(h) - f(0)}{h}$ to determine which case applies.