Consider four real functions, each defined on the whole real line, and examine differentiability at the stated point. Which conclusions hold?

(A) $g(x)=(x-1)(x+1)|x-1|$. Near $1$, $g(x)=\pm(x+1)(x-1)^2$, so $\frac{g(1+h)-g(1)}{h}=\pm(2+h)h\to0$ from both sides; the one-sided derivatives agree, so g is differentiable. True. (B) $\frac{h(x)-h(0)}{x}=\frac{|x|\cos x}{x}\to+1$ from the right and $-1$ from the left; false. (C) At $\pi/4$ the two functions cross with slopes $\cos\frac\pi4=\frac{1}{\sqrt2}$ and $-\sin\frac\pi4=-\frac1{\sqrt2}$, giving a corner; false. (D) The difference quotient is $\sin(1/h)$, which oscillates between $-1$ and $1$ and has no limit; false. Only A is correct. Remember the test: a function with a modulus factor is differentiable at the zero of the modulus only if the other factor also vanishes there, and the oscillating example fails because no limit exists.