Help with function accepting real values

Let me explain what went wrong.

When you define a function f[pattern] := ... then this will check at every occurrence of f[x] if x matches pattern.

Your pattern was x_Real which means "match any expression whose Head is Real". In Mathematica floating point numbers have the Head Real.

In[1]:= Head[0]
Out[1]= Integer

In[2]:= Head[0.]
Out[2]= Real

However, you do not care whether or not the argument has had Real, rather you want it to be an element of the set Reals. This is what Artes suggested in the comments by using a Condition

f[x_] /; Element[x, Reals] := x^2

Alternatively you could also use a PatternTest

f[x_?Element[#, Reals]&] := x^2

0 without the "." is an Integer, see

Head @ 0

(* Integer *)

but

f[0.]

delivers

0.