I have a function which does not have a closed form . Large numerical effort should be done to evaluate the function for even a single point. How can I examine the convexity of my function over the whole domain of function with minimum computational load? some extra assumption is that the function is continuous and almost every where differentiable!

If your function was strictly convex and $C^3$ (let's say on a compact interval $I$) and you had bounds on the third derivative there, it might be possible to prove its convexity on $I$ by evaluating it at sufficiently many points of $I$. If not, it's impossible to distinguish any convex function from one whose second derivative has a narrow "blip" taking it below zero between some of the points where you evaluated it. EDIT:
Here's what I had in mind. Suppose you know that $f''' \le B$ on interval
$I$. Evaluate $f$ at points $x_j = x_0 + j \delta$, $j = 0 \ldots n$ so that $I = [x_0, x_n]$, and compute $L_j = f(x_{j1})  2 f(x_{j}) + f(x_{j+1})$, $j = 1 \ldots n1$. If $f''$ is bounded away from $0$, this method is guaranteed to succeed
as long as $\delta$ is sufficiently small. 

