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Since the question is about "really contrived functional equations" you should define what do you count as "really contrived". There are some important classes of functional equations.

• Difference equations. They are discrete difference analogs of differential equations

• Iterative equations. They usually can be reduced to difference equations

• Delay differential equations. The combinations of difference and differential equations.

These classes are applied in many spheres. That's why they deserve to be researched. If you are speaking about equations that are outside of these classes, then indeed they are not frequently used in applied mathematics, but still may be interesting for research. Mathematicians research what they perceive as interesting, some research things that even in theory cannot be used in applied fields.

I also want to add some considerations about usefullness of solving completely random contrived equations. Such equations may lead to interesting insights into special functions and uncover interesting connections between them. Of course the mathematicians usually want to find general method which may be applicable not only to a particular equation but to a large class.

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Since the question is about "really contrived functional equations" you should define what do you count as "really contrived". There are some important classes of functional equations.

• Difference equations. They are discrete difference analogs of differential equations

• Iterative equations. They usually can be reduced to difference equations

• Delay differential equations. The combinations of difference and differential equations.

These classes are applied in many spheres. That's why they deserve to be researched. If you are speaking about equations that are outside of these classes, then indeed they are not frequently used in applied mathematics, but still may be interesting for research. Mathematicians research what they perceive as interesting, some research things that even in theory cannot be used in applied fields.