How do you find discontinuity of a piecewise function?

1 Answer
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.