# 5 d) Examine the following functions for continuity.  $f (x) = | x - 5|$

Given function is
$f (x) = | x - 5|$
for x > 5  , f(x) =  x - 5
for x < 5 , f(x) = 5 - x
SO, different cases are their
case(i)   x > 5
for every real number k > 5   , f(x) = x - 5 is defined
$f(k) = k - 5\\ \lim_{x\rightarrow k }f(x) = k -5\\ \lim_{x\rightarrow k }f(x) = f(k)$
Hence, function  f(x) = x - 5 is continous for x > 5

case (ii)      x < 5
for every real number k < 5   , f(x) = 5 - x is defined
$f(k) = 5-k\\ \lim_{x\rightarrow k }f(x) = 5 -k\\ \lim_{x\rightarrow k }f(x) = f(k)$
Hence, function  f(x) = 5 - x is continous for x < 5

case(iii)    x = 5
for x  = 5   , f(x) = x - 5 is defined
$f(5) = 5 - 5=0\\ \lim_{x\rightarrow 5 }f(x) = 5 -5=0\\ \lim_{x\rightarrow 5 }f(x) = f(5)$
Hence, function  f(x) = x - 5 is continous for x = 5

Hence, the function $f (x) = | x - 5|$ is continuous for each and every real number

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