Ohm's law and Joules heating Ohm's law state that the ratio of V and I gives us a constant value of R provided that the temperature is kept constant throughout. However, in accordance with the joule's heating it would get heated by Isquare. R. So, even if the I and V remains in a linear slope of the graph, should it be rendered as ohms law since that requires temperature to be constant.
 A: Ohm's law doesn't say anything about temperature. Ohm's law is analogous to Hooke's law in the sense that it is an idealization. Hooke's law idealizes a spring (or "spring-like" system) by assuming that the restoring force always is proportional to the displacement from equilibrium. Ohm's law idealizes resistors by assuming that the current through the resistor is always proportional to the voltage across the resistor.
Of course in the real world the heating of resistors can change their resistance, so we would say in those scenarios Ohm's law is no longer valid. This is analogous to "springs" that do not exactly follow a quadratic potential energy function, or perhaps even for springs that undergo deformation; we would say Hooke's law is no longer valid.
The above laws aren't laws in the same sense as say Newton's laws. They are better understood as idealizations.

So, even if the I and V remains in a linear slope of the graph, should it be rendered as ohms law...

Yes, I would say so.
A: The real "Ohms law" just defines what resistance is. $$R=\frac{U}{I}$$  and for real objects R can increase , decrease  or stay constant when I,U or T changes, thats why you get a characteristic curve for resistors, or you buy one whose R is constant over a certain range you need. Usually in school, Ohms law is U=R*I, since you don't vary U or I over a wide range.
