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  • Examine the following functions for continuity. f (x) is equal to | x minus 5|

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

 

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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 continuous 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 continuous 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 continuous for x = 5

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

Posted by

Gautam harsolia

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