f(x)={x if x∈Qx2 if x∈R−QTo show that f is continuous at 1. We must show that limx→1f(x)=f(1)Suppose f(x)=x limx→1f(x)=1Suppose f(x)=x2 limx→1f(x)=1 Since f(x)=1, then f is continuous at 1.
b.
Consider the neighborhood of 2(2−n1,2+n1)n∈NThen the neighborhood contain an irrational number xnn∈N. Since irrational number is dense in R.i.e.xn∈(2−n1,2+n1)⟹2−n1<xn<2+n1⟹−n1<xn−2<n1⟹∣xn−2∣<n1⟹xn→2 as n→∞Now, f(xn)=xn2 ∀n∈N as xn is irrational∴limn→∞f(xn)=limn→∞xn2=(limn→∞xn)2=22=4=f(2)=2Thus, f is discontinuous at 2
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