Analysis ? Continuity of a function at a point?

Question

Define f : R→R by f(x) = 3x^2 – 2x + 1. Show that f is continuous at 2.

Answer 1

take the limit as x approaches two. Show that it is the same from both sides. Then show that this is the same as the value at x=2

Answer 2

Just use the definition of continuity:
f(x) is continuous at 2 if for all ε>0 there exists δ>0 such that for all x with |x-2|<δ, |f(x)-f(2)|<ε.

Lets suppose we are given ε>0.
Now pretend we have chosen some suitable δ - we'll come back to this later, and assume |x-2|<δ.
We want to prove that |f(x)-f(2)| < δ.
|f(x) - f(2)| = |3x^2 - 2x + 1 - 3*2^2 + 2*2 - 1| = |3(x^2 - 2^2) - 2(x-2)|.
= |3(x+2)(x-2) - 2(x-2)|
= |x-2||(3(x+2) - 2)|
< δ|3x+4|
= δ|3(x-2) + 10|
<= δ(|3(x-2)| + 10)
< δ(3δ+10).

Thus all we have to do is choose δ originally such that δ(3δ+10) < ε, and we're done.

Thats easy; I'll leave the algebra of it to you.