Consumption smoothing in RBC Model I have the following inter temporal utility function: 

$U(t)=(\frac{s(t)}{1-\sigma})(c_t/c_{t-1}^\gamma)^{(1-\sigma)} - \chi*h(t)$

where $h(t)$ is the hours worked.
I know that gamma is responsible for consumption smoothing as I have achieved the following results after a positive shock on $s(t)$:

The problem is i do not know how to prove mathematically that when gamma=1 consumption is smoother
Have you tried solving the maximization problem? Normally you do this taking into account some market clearing condition or budget constraint and you should be able to solve for equilibrium prices to get future consumption as a function of present consumption. Consumption smoothing will simply show the path of consumption. 

Getting this equation you can show how consumption varies for different levels of $ \gamma $ (in your case = 0 and = 1)