MathLabs

Problem 2

Find all functions f:R→Rf:\mathbb R\to\mathbb R such that f(x2+f(y))=y+f(x)2f(x^2+f(y))=y+f(x)^2 for all real x,yx,y.
Step 4 of 5: Extend additivity to all real increments
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In plain words

Extend additivity to all real increments

f(−x)=−f(x),f(x+y)=f(x)+f(y)f(-x)=-f(x),\qquad f(x+y)=f(x)+f(y)
Detailed analysis

Putting y=−xy=-x in the positive-increment identity yields f(−x)=−f(x)f(-x)=-f(x) for x>0x>0. This oddness, together with the identity for positive increments, gives f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y) and f(x−y)=f(x)−f(y)f(x-y)=f(x)-f(y) for all real x,yx,y.