Continuity

54 problems · Continuous functions and their main theorems.

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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.4Continuity with parameters

#continuity #piecewise functions #parameters

Continuous and discontinuous functions

High school

Ex.6Extending a rational function by continuity

#extension by continuity #factoring #removable singularity

Continuous and discontinuous functions

High school

Ex.16f(x)f(y) − f(xy) = x + y

#substitution #special values #verification

Functional equations

Ex.25A fixed point for every continuous self-map of [0,1]

#fixed points #intermediate value theorem

Intermediate Value Theorem

High school

Ex.26Extensions by continuity and oscillation

#extension by continuity #oscillation #squeeze theorem

Continuous and discontinuous functions

Ex.27Locating the roots of a cubic

#intermediate value theorem #polynomial roots #bracketing

Intermediate Value Theorem

High school

Ex.28How long is the drug effective?

#applications #intermediate value theorem #bracketing

Intermediate Value Theorem

High school

Ex.30Where is 1/ln|x| continuous?

#continuity #limits #logarithm

Continuous and discontinuous functions

Ex.31Which of these functions extend by continuity?

#extension by continuity #one-sided limits #exponential

Continuous and discontinuous functions

Ex.54Floor plus square root of the fractional part

#continuity #floor function #one-sided limits

Continuous and discontinuous functions

Ex.55The monk on the mountain

#intermediate value theorem #classic puzzle

Intermediate Value Theorem

High school

Ex.70The midpoint functional equation

#functional equations #affine functions

Functional equations

Ex.78Glued along the rationals and the irrationals

#continuity #density #sequences #pathological functions

Continuous and discontinuous functions

Ex.79The limit of an iterated sequence is a fixed point

#sequences #fixed points #continuity #golden ratio

Continuous and discontinuous functions

Ex.80Maximum and minimum of two continuous functions

#continuity #absolute value #algebra of continuous functions

Continuous and discontinuous functions

Ex.81The absolute value of a continuous function

#continuity #absolute value #counterexamples

Continuous and discontinuous functions

Ex.87Fixed point on [0,1]

#fixed point #IVT #auxiliary function

Intermediate Value Theorem

High school

Ex.88A continuous periodic function is bounded

#extreme value theorem #periodic functions #boundedness

Continuous and discontinuous functions

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

Olympiad

Ex.100f(x + y) + f(x − y) = 2f(x)f(y)

#d'Alembert #cosine #differential equation #differentiability

Functional equations

Olympiad

Ex.101f(x² + f(y)) = y + f(x)²

#injectivity #surjectivity #oddness #Cauchy equation

Functional equations

Olympiad

Ex.105Commuting maps on $[0,1]$ share a fixed point

#commuting maps #IVT #extreme value theorem #induction

Intermediate Value Theorem

Ex.106Lipschitz functions are uniformly continuous

#Lipschitz #uniform continuity #epsilon-delta

Continuous and discontinuous functions

Ex.110Continuous functions with f^2=1 on an interval

#IVT #sign #constant functions

Intermediate Value Theorem

High school

Ex.1111/x on (0,1] is not uniformly continuous

#uniform continuity #open intervals #Heine theorem

Continuous and discontinuous functions

Ex.112The square root: uniformly continuous, not Lipschitz

#uniform continuity #Lipschitz #square root

Continuous and discontinuous functions

Ex.113x² is not uniformly continuous on the real line

#uniform continuity #sequences #counterexamples

Continuous and discontinuous functions

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

#functional equation #cosine #continuity #monotonicity

Functional equations

Ex.316A cubic with exactly one real root

#intermediate value theorem #monotonicity

Intermediate Value Theorem

Ex.354A continuous function that vanishes on the rationals

#density #continuity #sequences

Continuous and discontinuous functions

Ex.360Odd degree polynomials have a real root

#intermediate value theorem #polynomials #limits at infinity

Intermediate Value Theorem

Ex.392The bisection method in practice

#bisection #numerical methods #intermediate value theorem

Intermediate Value Theorem

Ex.639Cauchy's equation on the rationals

#Cauchy equation #additive function #rationals

Functional equations

Ex.720A continuous function with rational values is constant

#intermediate value theorem #rational numbers #proof by contradiction

Intermediate Value Theorem

Ex.736cos x = 1/x has infinitely many solutions

#intermediate value theorem #infinitely many solutions

Intermediate Value Theorem

Ex.737Counting solutions depending on a parameter

#intermediate value theorem #monotonicity #parameters

Intermediate Value Theorem

Ex.756tan x = x on each branch

#intermediate value theorem #tangent #monotonicity

Intermediate Value Theorem

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

#injectivity #involution #Cauchy equation on Q

Functional equations

Olympiad

Ex.1314f(x)f(y) − f(xy) = x + y

#functional equation #substitution #polynomial identity

Functional equations

Ex.1444Continuity of a piecewise function with an essential singularity

#piecewise #continuity #exponential singularity

Continuity

Ex.1479Continuous injective functions are monotone

#intermediate value theorem #injectivity #monotonicity

Intermediate Value Theorem

Ex.1491How many solutions does ln x = ax have?

#intermediate value theorem #parameters #logarithm

Intermediate Value Theorem

Ex.1512The horizontal chord of length one half

#intermediate value theorem #chords #clever auxiliary function

Intermediate Value Theorem

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

#IMO #natural numbers #induction #monotonicity

Functional equations

Olympiad

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

#additive function #monotonicity #Cauchy equation

Functional equations

Olympiad

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

#functional equation #integers #parity #recursion

Functional equations

Olympiad

Ex.1981f(f(x) + y) = 2x + f(f(y) − x)

#surjectivity #substitution #linear functions

Functional equations

Olympiad

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

#IMO 1968 #periodicity #squaring #completing squares

Functional equations

Olympiad

Ex.1983IMO 2010: f(⌊x⌋y) = f(x)⌊f(y)⌋

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

Functional equations

Olympiad

Ex.2080The popcorn function

#continuity #rationals #density #pathological functions

Continuous and discontinuous functions

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

#functional equation #monotone #integers #ternary structure

Functional equations

Olympiad