1. The problem discusses the concept of the derivative of a function $f$ at a point $a$ from the right and from the left. 2. The right-hand derivative at $a$ is defined as $$f'(a^+) = \lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h}.$$ This limit considers values of $h$ approaching zero from the positive side. 3. The left-hand derivative at $a$ is defined as $$f'(a^-) = \lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h}.$$ This limit considers values of $h$ approaching zero from the negative side. 4. For the derivative $f'(a)$ to exist (be defined), these two one-sided derivatives must be equal: $$f'(a^+) = f'(a^-).$$ 5. If $$f'(a^+) \neq f'(a^-),$$ then the derivative at $a$ is undefined because the slope of the tangent line from the right and left do not match. 6. This situation often occurs at points where the function has a sharp corner or cusp. 7. Therefore, the derivative $f'(a)$ is undefined if the right-hand and left-hand derivatives at $a$ are not equal.