Problem

Suppose that $f$ is a differentiable function on $(-\infty,\infty)$ such that

$$f(x)=ax^2+bx+c \quad \text{for }x\in(-1,0)$$

for some constants $a,b,c$ and

$$f(x)=-0.5 \quad \text{for }x<-1.$$

Which of the following statements is false?

(a) $c=a-0.5$.

(b) If $f(0)=0$, then $f'(0)=2$.

(c) If $f'(0)=2$, then $a=1$.

(d) If $f(0)=0$, then $f(x)+f(-x)=0$ for $x\in(-1,1)$.

(e) $f$ is continuous at $0$.