Question

Examine the following functions for continuity. (a) (b) (c) (d)

Answer 1

(a)

The given function is,

f( x )=x−5

It is a polynomial function. So, f is defined for all real numbers of c.

The limit of the function is,

lim x→c f( x )= lim x→c x−5 =c−5 =f( c )

Thus, f is continuous for all real numbers. So, it is a continuous function.

(b)

Given function is,

f( x )= 1 x−5

Calculate the value of the function at x=5,

f( x )= 1 5−5 = 1 0 =∞

So, at x=5 function f( x ) is not defined.

Therefore, f is defined for all real numbers of c except 5.

Limit of the function is,

lim x→c f( x )= lim x→c 1 x−5 = 1 c−5

Also,

f( c )= 1 c−5

Therefore, lim x→c f( x )=f( c )

Thus, f is continuous for all real numbers except 5. So, function is continuous function.

(c)

Given function is,

f( x )= x 2 −25 x+5

Calculate the value of the function at x=−5,

f( x )= 25−25 −5+5 = 0 0 =Undefined

So, at x=−5 function f( x ) is not defined.

Therefore, f is defined for all real number c.

Given function is continuous at x=c, here c is a real number except −5.

Limit of the function is,

lim x→c f( x )= lim x→c x 2 −25 x+5 = c 2 −25 c+5 = ( c−5 )( c+5 ) c+5 =c−5

Also,

f( c )= c 2 −25 c+5 = ( c−5 )( c+5 ) c+5 =c−5

Therefore, lim x→c f( x )=f( c )

Thus, f is continuous for all real numbers except −5. So, function is continuous.

(d)

Given function is,

f( x )=| x−5 |

Case 1:

Check the continuity of the function at x=5.

f( c )=f( 5 ) =5−5 =0

Left hand limit of the function is,

lim x→ 5 − f( x )= lim x→ 5 − | x−5 | = lim x→ 5 − −( x−5 ) =−( 5−5 ) =0

Right hand limit of the function is,

lim x→ 5 − f( x )= lim x→ 5 + | x−5 | = lim x→ 5 + ( x−5 ) =( 5−5 ) =0

Therefore, LHL=RHL=f( 5 )

Therefore, function is continuous at x=5.

Case 2:

Check continuity at x=c where c>5,

f( x )=| x−5 | =( x−5 )

The limit of the function is,

lim x→c f( x )= lim x→c x−5 =c−5

Also,

f( c )=c−5

Therefore, lim x→c f( x )=f( c )

Function is continuous at each x=c, c>5.

Case 3:

Check continuity at x=c where c<5,

f( x )=| x−5 | =−( x−5 )

The limit of the function is,

lim x→c f( x )= lim x→c x−5 =−( c−5 ) =−c+5

Also,

f( c )=−( c−5 ) =−c+5

Therefore, lim x→c f( x )=f( c )

Thus, function is continuous at each x=c, c<5.

Hence, f( x )=| x−5 | is continuous at every real number. So, it is a continuous function.