How do you find discontinuity of a piecewise function?

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.