Question

How do you find discontinuity of a piecewise function?

Answer 1

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.