Validating a function over a domain in a computer program.

I am working on the front end of an engineering software, where I have an evaluator to return the value of a function at a point in a given domain, lets call it eval(f(x)) where f(x) is itself single parameter mathematical function, or a combination of such functions, and x is any real number.

I need to validate f(x) only over a continuous subset of the domain, and if the range of f(x) in the given domain is out of bounds, I am supposed to return an error.

I also have a given domain this function is to work over, and the range this function is supposed to spew results over. eval() itself returns certain error messages, such as 0 given inside a log function, (My backend can handle this at the domain's end point, so I need my frontend to not return an error when this case hits)

I basically want to know what individual values of the continuous domain I should evaluate my values at, and how do I evaluate my function in an exclusive range(so I can exclude the endpoints of my domain), what step size should I use to reasonably divide my given domain into values?

Keep in mind, I have no other mathematical operators apart from eval() I can use on my function, and I don't want to make my own for the sake of computer power, unless absolutely needed.

Extra info about the function: The f(x) is a standard or a combination of standard functions, it is never a piece-wise defined function. Also, I only care about the rules I have of my own about the function: (e.g. is log(9999)>2222) rather than if log(0) is valid or not). I can also make do with a check for the range being a continuous range, and not having infinities or any close values in it.

I also don't expect my users to enter functions which are indeterminate, I just want to check the validity of the range of the entered functions(Against my own rules) rather than the correctness of the function itself, in case the function is invalid, I actually want to ignore that since eval() already gives errors for those conditions(Apart from f(x) domain endpoints, which I need to ignore specifically)

I don't really expect f'(x) to have a very high value either(unless its going towards infinity(e.g. -log(x-1) at 1;

Bonus question:

In the future, my f(x) also might become f(x,y), with a similar restrictions on the domain of y, any advice on that?

Given some knowledge about the bounds of $f'(x),$ you could use a finite number of observations of $f(x)$ to conclude that $f(x)$ was within certain bounds as long as $x$ is within certain bounds, but if $f(x)$ is just some arbitrary function, it could be zero everywhere except when $1.500001 < x < 1.500002$ and we could have $f(1.5000015) = 10^{100}.$ That's an extreme example, but it does illustrate that you need to know something more than simply "I have a function" before you can decide on a suitable step size.
You can estimate $f'(x)$ by evaluating $f(x)$ at slightly different values of $x,$ but as the previous example illustrated, there's no guarantee that any particular set of values you used when estimating $f'(x)$ will tell you how large $f'(x)$ can actually get.
In reality, you probably do have some kind of information like this about $f(x),$ even if it's very fuzzy information (such as, "The engineers wouldn't give me a function with such a large derivative as the one in the example").