Maths WorkshopMaths problems with full solutions

Functional equations

19 problems · Cauchy-type functional equations.

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Ex.1A functional equation with f(2x)

#functional equations #continuity at a point #sequences

Functional equations

Ex.3Cauchy's equation f(x + y) = f(x) + f(y)

#functional equations #density #rational numbers

Functional equations

Ex.16f(x)f(y) − f(xy) = x + y

#substitution #special values #verification

Functional equations

Ex.70The midpoint functional equation

#functional equations #affine functions

Functional equations

Ex.93Additive and multiplicative functions

#Cauchy equation #squares are nonnegative #monotonicity

Functional equations

Ex.94Cauchy's equation with monotonicity

#Cauchy equation #rationals #density

Functional equations

Ex.100f(x + y) + f(x − y) = 2f(x)f(y)

#d'Alembert #cosine #differential equation #differentiability

Functional equations

Ex.101f(x² + f(y)) = y + f(x)²

#injectivity #surjectivity #oddness #Cauchy equation

Functional equations

Ex.114A continuous f with f o f= cos cannot exist

#functional equation #cosine #continuity #monotonicity

Functional equations

Ex.639Cauchy's equation on the rationals

#Cauchy equation #additive function #rationals

Functional equations

Ex.1258f(x + f(y)) = f(x) + y on the rationals

#injectivity #involution #Cauchy equation on Q

Functional equations

Ex.1314f(x)f(y) − f(xy) = x + y

#functional equation #substitution #polynomial identity

Functional equations

Ex.1943An inequality as a functional equation (IMO 1977)

#IMO #natural numbers #induction #monotonicity

Functional equations

Ex.1979Additive functions with f(x²) = f(x)²

#additive function #monotonicity #Cauchy equation

Functional equations

Ex.1980Solving f(f(n)) = n + 2 on the positive integers

#functional equation #integers #parity #recursion

Functional equations

Ex.1981f(f(x) + y) = 2x + f(f(y) − x)

#surjectivity #substitution #linear functions

Functional equations

Ex.1982IMO 1968: f(x + a) = ½ + √(f(x) − f(x)²)

#IMO 1968 #periodicity #squaring #completing squares

Functional equations

Ex.1983IMO 2010: f(⌊x⌋y) = f(x)⌊f(y)⌋

#IMO 2010 #floor function #functional equation #case analysis

Functional equations

Ex.2238A strictly increasing f with f(f(n)) = 3n

#functional equation #monotone #integers #ternary structure

Functional equations