In most cases, we should look for a discontinuity at the point where a piecewise defined function c

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.