How do you find discontinuity of a piecewise function?

1 Answer
Write your answer here...
Start with a one sentence answer
Then teach the underlying concepts
Don't copy without citing sources
preview
?

Answer

Write a one sentence answer...

Answer:

Explanation

Explain in detail...

Explanation:

I want someone to double check my answer

Describe your changes (optional) 200

35
Wataru Share
Oct 3, 2014

In most cases, we should look for a discontinuity at the point where a piecewise defined function changes its formula. You will have to take one-sided limits separately since different formulas will apply depending on from which side you are approaching the point. Here is an example.

Let us examine where #f# has a discontinuity.

#f(x)={(x^2 if x<1),(x if 1 le x < 2),(2x-1 if 2 le x):}#,

Notice that each piece is a polynomial function, so they are continuous by themselves.

Let us see if #f# has a discontinuity #x=1#.

#lim_{x to 1^-}f(x)=lim_{x to 1^-}x^2=(1)^2=1#

#lim_{x to 1^+}f(x)=lim_{x to 1^+}x=1#

Since both limits give 1, #lim_{x to 1}f(x)=1#

#f(1)=1#

Since #lim_{x to 1}f(x)=f(1)#, there is no discontinuity at #x=1#.

Let us see if #f# has a discontinuity at #x=2#.

#lim_{x to 2^-}f(x)=lim_{x to 2^-}x=2#

#lim_{x to 2^+}f(x)=lim_{x to 2^+}(2x-1)=2(2)-1=3#

Since the limits above are different, #lim_{x to 2}f(x)# does not exist.

Hence, there is a jump discontinuity at #x=2#.

I hope that this was helpful.

Was this helpful? Let the contributor know!
1500
Impact of this question
16161 views around the world
You can reuse this answer
Creative Commons License