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continuous

Source: Romania District Olympiad 2013,grade XI(problem 4)

March 14, 2013

functionlimitcalculuscalculus computations

Problem Statement

Let

f:\mathbb{R}\to \mathbb{R}

be a monotone function. a) Prove that

f

have side limits in each point

{{x}_{0}}\in \mathbb{R}

. b) We define the function

g:\mathbb{R}\to \mathbb{R}

,

g\left( x \right)=\underset{t\nearrow x}{\mathop{\lim }}\,f\left( t \right)

(

g\left( x \right)

with limit at at left in

x

). Prove that if the

g

function is continuous, than the function

f

is continuous.

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