Maths WorkshopMaths problems with full solutions

Continuous and discontinuous functions

17 problems · Examples of continuous and discontinuous functions.

Ex.4Continuity with parameters
#continuity #piecewise functions #parameters
Continuous and discontinuous functions
High school
High school
Ex.6Extending a rational function by continuity
#extension by continuity #factoring #removable singularity
Continuous and discontinuous functions
High school
High school
Ex.26Extensions by continuity and oscillation
#extension by continuity #oscillation #squeeze theorem
Continuous and discontinuous functions
L1
L1
Ex.30Where is 1/ln|x| continuous?
#continuity #limits #logarithm
Continuous and discontinuous functions
L1
L1
Ex.31Which of these functions extend by continuity?
#extension by continuity #one-sided limits #exponential
Continuous and discontinuous functions
L1
L1
Ex.54Floor plus square root of the fractional part
#continuity #floor function #one-sided limits
Continuous and discontinuous functions
L1
L1
Ex.78Glued along the rationals and the irrationals
#continuity #density #sequences #pathological functions
Continuous and discontinuous functions
L1
L1
Ex.79The limit of an iterated sequence is a fixed point
#sequences #fixed points #continuity #golden ratio
Continuous and discontinuous functions
L1
L1
Ex.80Maximum and minimum of two continuous functions
#continuity #absolute value #algebra of continuous functions
Continuous and discontinuous functions
L1
L1
Ex.81The absolute value of a continuous function
#continuity #absolute value #counterexamples
Continuous and discontinuous functions
L1
L1
Ex.88A continuous periodic function is bounded
#extreme value theorem #periodic functions #boundedness
Continuous and discontinuous functions
L1
L1
Ex.106Lipschitz functions are uniformly continuous
#Lipschitz #uniform continuity #epsilon-delta
Continuous and discontinuous functions
L1
L1
Ex.1111/x on (0,1] is not uniformly continuous
#uniform continuity #open intervals #Heine theorem
Continuous and discontinuous functions
L1
L1
Ex.112The square root: uniformly continuous, not Lipschitz
#uniform continuity #Lipschitz #square root
Continuous and discontinuous functions
L1
L1
Ex.113x² is not uniformly continuous on the real line
#uniform continuity #sequences #counterexamples
Continuous and discontinuous functions
L1
L1
Ex.354A continuous function that vanishes on the rationals
#density #continuity #sequences
Continuous and discontinuous functions
L1
L1
Ex.2080The popcorn function
#continuity #rationals #density #pathological functions
Continuous and discontinuous functions
L2
L2