Trap 1: f' exists implies f' continuous FALSE. $x^2$ sin$\frac{1}{x}$ has f'(0) = 0 but f' oscillates near 0.

Trap 2: |f| continuous implies f continuous FALSE. f = 1 for rational, -1 for irrational. |f| = 1 (continuous) but f is nowhere continuous.

Trap 3: lim f'(x) = f'(a) Not necessarily. f'(a) is defined by the limit of the difference quotient, not the limit of f'. These agree if f' is continuous (L'Hopital-like result), but can differ.

Trap 4: Degree of non-differentiability of sums |x| + |x-1|: 2 non-differentiable points. But |x| + |x| = 2|x|: still 1 non-differentiable point (corners at the same location reinforce, not cancel).

Trap 5: f(x) = [x] + [-x] Not equal to 0! For non-integer x: [x] + [-x] = -1. For integer x: [x] + [-x] = 0.

Trap 6: Differentiability of |$x^n$| |x| not differentiable at 0. |$x^2$| = $x^2$ differentiable. |$x^3$| = |x|^3 differentiable. Rule: |$x^n$| differentiable at 0 iff n >= 2.