this is a very general question, and I might have missed out some conditions in order for the above statement in question to be true. If I have, please help me state these conditions, thank you.
It is often used in derivations in physics, when you have an equation, take for example in thermodynamics, we have the equation relating two systems:
$$\left(\frac{\partial S}{\partial E_1}\right)_{V_{1}, N_{1}} = \left(\frac{\partial S}{\partial E_2}\right)_{V_{2}, N_{2}}$$
The derivation goes on to state, since the two sides of the equation are equal, they must be equal to a constant, thus the constant, temperature $T$ is defined to be:
$$\left(\frac{\partial S}{\partial E_i}\right)_{V_{i}, N_{i}} \equiv \frac{1}{T_i}$$
This kind of thinking is used as well in the derivation of Schrondinger's equation. Could anyone explain to me what the mathematical reason is? As well as any conditions I might have left out.