Is it true that $f(a,b)=f(a,c)$ whenever $b=c$? How to prove this (true or not)?

$f(a,b) = f(a,c)$  must hold if $b = c$

Note: f(a,b) is a function with a & b parameters
thanks
 A: Hint: by definition, a function takes in some inputs, and produces a unique output.
A: See it is a bi variate function that you have given here.
Just think of a 3-D(2 dimentional system of co-ordinate).
Now if a function is a bivariate one then the functional value will have the value on axis which one is mutually perpendicular to the other two axis.
Now you have the equation as f(a,b)=f(a,c).Now we have to prove that the equation can only hold if b=c.
But see can we say anythhing about the functional characteristics like if f(a,b)=f(a,c) then 'b' should be equal to 'c'!
How can anyone conclude it surely without knowing the actual function?
See here I have an example:  f(a,b)= (a^2)+(b^2)  and  f(a,c)= (a^2)+(c^2).
See here,for the first function f(a,b) if we put a=1 and b=1 then the functional value will be 2,fine.If we put in the 2nd function a=1 and c=(-1) then also the functional value is 2 but see here for these two function b(b=1) is not equal to c(c=-1) for these two cases but still the functional value are same!!
So what do you think my friend.Don't we need the actual characteristics of the function?  
A: If we assume that $\forall$x(x=x), then yes.
Suppose that b=c.
f(a,b)=f(a,b) since we can derive it from substitution and that $\forall$x(x=x).
Now since b=c we can replace just the second "b" by "c" and obtain
f(a,b)=f(a,c).
