Question

use the definition of continuity to prove that f(x) is continuous at x=0 for f(x) = {2x+x^2sin(1/x) if x not= 0, 0 if x=0}

Answer 1

if f(x) is continuous at 0, then when f(x) is greater or less than 0 or when f(x )=0, we have to find the same answer.
when f(x) is not equal to 0, that is f(x) is less or greater than 0, we should use f(x)=2x+x^2sin(1/x). put 0 instead of x. sin(1/x) as x goes to 0 does not exists but does not matter. you would find 0.
when f(x) is equal to 0, the answer is already determined in the question 0.
thus f(x) is continuous at 0