Limits, Continuity and Differentiability Part 1

Let a is a rational number other than 0 , in
[−5,5], then
f(a)=a and

lim

x→a

f(x)=−a
[ As in the immediate neighborhood of a rational - number, we find irrational numbers ]
∴f(x) is not continuous at any rational number
If a is irrational number, then
f(a)=−a and

lim

x→a

f(x)=a
∴f(x) is not continuous at any irrational number clearly

lim

x→0

f(x)=f(0)=0
∴f(x) is continuous at x=0