Question Number 72634 by Rio Michael last updated on 30/Oct/19
$${help}\:{me}\:{with}\:{the}\:{conditions}\:{please}\: \\ $$$${for}\:{a}\:{function}\:{f}\:{to}\:{be}\:{continuous}\:{at}\:{a}\:{point}\:{a} \\ $$
Commented by Prithwish sen last updated on 31/Oct/19
$$\boldsymbol{\mathrm{If}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{can}}\:\boldsymbol{\mathrm{able}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{draw}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{function}}\:\boldsymbol{\mathrm{f}}\:\:\boldsymbol{\mathrm{through}} \\ $$$$\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{point}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{without}}\:\boldsymbol{\mathrm{lifting}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{pen}}\:\boldsymbol{\mathrm{then}}\:\boldsymbol{\mathrm{f}}\:\boldsymbol{\mathrm{is}} \\ $$$$\boldsymbol{\mathrm{continuous}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{a}} \\ $$
Commented by Rio Michael last updated on 31/Oct/19
$${thanks}\:{sir}\:{hahah} \\ $$
Commented by Prithwish sen last updated on 31/Oct/19
$$\mathrm{welcome} \\ $$
Commented by mathmax by abdo last updated on 31/Oct/19
$${f}\:{continue}\:{at}\:{x}_{\mathrm{0}} \Leftrightarrow{lim}_{{x}\rightarrow{x}_{\mathrm{0}} } \:\:{f}\left({x}\right)={f}\left({x}_{\mathrm{0}} \right) \\ $$
Answered by mind is power last updated on 31/Oct/19
$$\:\mathrm{f}\:\mathrm{is}\:\mathrm{continus}\:\mathrm{in}\:\mathrm{a}\:\mathrm{if}\:,\forall\varepsilon>\mathrm{0}\:\exists\eta>\mathrm{0}\:\:\forall\mathrm{x}\:\:\mid\mathrm{x}−\mathrm{a}\mid<\eta\Rightarrow\mid\mathrm{f}\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{a}\right)\mid<\epsilon \\ $$$$\mathrm{What}\:\mathrm{do}\:\mathrm{you}\:\mathrm{dont}\:\mathrm{understand}\:\mathrm{sir}\:? \\ $$
Commented by Rio Michael last updated on 31/Oct/19
$${yeah}\:{sir}\:{i}\:{don}'{t} \\ $$