We see that and also that.
Propertiesof ContinuousFunctions
Let’s recallour exampleof the limit of compositefunctions:
We saw thatf(g(x)) is undefinedand has the indeterminateform of 1/0. Hence doesnot
exist.
In general,we will requirethatfbe continuousatx = g(a) andx=g(a) mustbe in the domainof (f o g) in
orderfor to exist.
We will statethe followingtheoremand delayits proofuntil Chapter3 whenwe havelearnedmoreabout
real numbers.
Min-MaxTheorem: If a functionf(x) is continuousin a closedintervalI, thenf(x) has both a maximumvalue
and a minimumvalueinI.
Example2:
Consider and interval
The functionhas a minimumvalueat valueat and a maximumvalueat
wheref(2) = 9