Consider the following. f ( x ) = Describe the interval(s) on which the function

Question

Consider the following.

f(x) =

Describe the interval(s) on which the function is continuous. (Enter your answer using interval notation.)
( )

Identify any discontinuities. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

x =

If the function has a discontinuity, identify the conditions of continuity that are not satisfied.

There is a discontinuity at x = c where f(c) is not defined.

There is a discontinuity at x = c where lim xc f(x) does not exist.

There are no discontinuities; f(x) is continuous.

There is a discontinuity at x = c where lim xc f(x) f(c).

Explanation / Answer

For the function f(x) to be continuous at x = c

lim [x -> c-] f(x) = lim [x -> c+] f(x) = f(c)

both 5x +1 , x2 -49 are continuous functions

Checking continuity at x = 0

==> lim [x -> 0-] f(x) = lim [x -> 0-] x2 -49          since for values less than 0 , f(x) = x2 -49

==> 02 -49 = -49

==> lim [x -> 0+] f(x) = lim [x -> 0+] 5x +1      since for values greater than 0 , f(x) = 5x +1

==> 5(0) +1 = 1

as lim [x -> 0-] f(x) lim [x -> 0+] f(x)

function is discontinuous at x = 0

There is a discontinuity at x = c where lim xc f(x) does not exist ; here c = 0

continuous in the interval (- , 0) U (0 , )

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