f(x) and g(x) are functions defined on $R\to R$
Statement 1: If f(x) is continuous and g(x) is discontinuous at x = a then their product is continuous at x = a if f(a) = 0
Statement 2: If f(x) is continuous and g(x) is discontinuous at x = a and if g(a) = 0 then their product is continuous at x = a
Statement 3: If f(x) is differentiable and g(x) is not derivable at x = a then their product is derivable at x = a if g(x) is continuous and f(a) = 0
Statement 4: If f(x) is differentiable and g(x) is not derivable at x = a then their product is derivable at x = a if g(x) is continuous and g(a) = 0
Which of the following must be truth value of above statements in that order.

Statement 1

 If f(x) is continuous and g(x) is discontinuous at x = a then their product is continuous at x = a if f(a) = 0

The above statement is false> 

Suppose that $f\left(x\right)=x,g\left(x\right)=\frac{\left|x\right|}{x}$, here the function $f\left(x\right)$ is continuous at $x=0$ and $g\left(x\right)$ is discontious at $x=0$ but $f\left(x\right)g\left(x\right)$ is not continous at $x=0$

-Statement II :  If f(x) is continuous and g(x) is discontinuous at x = a and if g(a) = 0 then their product is continuous at x = a

The above statement is False

$f\left(x\right)=x$ and $$\begin{gathered} g\left(x\right)=\left\{\begin{array}{l}\frac{\left|x\right|}{x^2} x\ne 0 \\ \\ =0 x=0\end{array}\right. \end{gathered}$$ is discontinuous at $x=0$ and $g\left(0\right)=0$

The product of the functions is not continuous at $x=0$ 

-Statement III :If f(x) is differentiable and g(x) is not derivable at x = a then their product is derivable at x = a if g(x) is continuous and f(a) = 0

Yes the above statement is true. 

Statement 4: If f(x) is differentiable and g(x) is not derivable at x = a then their product is derivable at x = a if g(x) is continuous and g(a) = 0

The above statement is False.