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3.6 Properties Of Functions Continuous On An Interval

Property 1: If a function f is continuous on [ a, b ], then the graph of the function is unbroken ( i.e. continuous) curve joining the points (a, f (a) ) and (b, f ( b))

Property 2: If f is the continuous on [ a, b ] and f ( c ) ¹ 0 and a < c < b, then there exists a small positive number d. So that f ( x ) is of the same sign as f ( c ) then c - d < x < c + d. i.e. a continuous function does not undergo sudden changes. This property will be later on used for the discussion of Maxima and Minima.

Property 3: If f is continuous on [a , b] and f (a), f (b) are opposite signs, then f (x) = 0 for at least one value of x in (a , b).

 

Property 4: If f is continuous on a closed interval [a , b] then

i) f is bounded in [ a, b ]

ii) There exists points c and d in [ a, b] where f assumes its lub, and glb, M and m respectively.

i.e. f (c) = M and f (d) = m

iii) f assumes every value between m and M at least one value x [ a, b ]

Note :

1) A discontinuous function may be bounded but may not attain the glb or lub.

Even a continuous function on an open interval may not attain glb and lub.

2) These properties are just the necessary conditions but not the sufficient condition to make a function continuous.

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[next chapter]

 

Index

3.1 Continuity At a Point
3.2 Continuity In An Interval
3.3 Some Very -often - encountered Continuous Functions
3.4 Algebra Of Continuous Functions
3.5 Discontinuity And its Classification
3.6 Properties of Functions Continuous on an Interval

Chapter 4





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