Define two functions:
functionWithConditional[t_] := If[t < 0, 0, t]
functionWithSign[t_] := (Sign[t] + 1)*t/2
Mathematica considers them equal in this sense:
In[3]:= FullSimplify[functionWithConditional[t] - functionWithSign[t]]
Out[3]= 0
But the difference between their transforms is nonzero:
In[5]:= FourierTransform[functionWithConditional[t], t, s] -
FourierTransform[functionWithSign[t], t, s]
Out[5]= I Sqrt[\[Pi]/2] Derivative[1][DiracDelta][s]
And the inverse-transform of the difference is non-zero:
In[6]:= InverseFourierTransform[%, s, t]
Out[6]= -(t/2)
Which if these results is correct, and which is incorrect:
In[7]:= FourierTransform[functionWithConditional[t], t, s]
Out[7]= -(1/(Sqrt[2 \[Pi]] s^2))
In[8]:= FourierTransform[functionWithSign[t], t, s]
Out[8]= -(1/(Sqrt[2 \[Pi]] s^2)) - I Sqrt[\[Pi]/2] Derivative[1][DiracDelta][s]
Or are they somehow both correct, despite Mathematica claiming the functions are equal? Why should two equal functions have unequal transforms?