Differentiating piecewise functions. Say we have the piecewise function $f(x) = x^2$ on the interval $0 \le x < 4$; and it equals $x+1$ on the interval $ x \ge 4$. Why is it that, when I take the derivative, the intervals loose their equality and become strictly greater or strictly less than?


You see how the function jumps at $x=4$?  It isn't even continuous, so it makes no sense to talk about it being differentiable there.  $f'(4)$ is simply indeterminate, though $f'(x)$ is perfectly easy to define on $\{x:x\in \mathbb{R}\setminus \{4\}|x>0\}$.
