T/F (feat. symmetry) #2

Which of the following is ALWAYS TRUE ?

IF TRUE prove it, IF FALSE find a way to fix it.

Note: All the functions below are continuous functions defined in all reals. Also 1. 2. 3. are independent problems.

(1). $f(x)+f(-x)=0 \iff \displaystyle \int_{-x}^x f(t) \,dt=0$

(2). $\displaystyle \bigg \{x \bigg | \int_{g(x)}^{h(x)} f(t)\,dt=0 \bigg \}=\{x|g(x)+h(x)=2a\}$ , where $g(x)<h(x)$ and $f(x)$ is an increasing function satisfying $f(x)+f(2a-x)=0$ .

(3). For non-negative integers $n$ , $\displaystyle \bigg \{x \bigg | \int_{g(x)}^{h(x)} f(t)\,dt=0\bigg \}= \big \{x|g(x)+h(x)=2a+4n(b-a)\big \}$ , where $g(x)<h(x)$ , $f(x)+f(2a-x)=0$ and $f(x)-f(2b-x)=0$ .