Use the definition of continuity to prove that the function f defined by f(x)=x^(1/2) is continuous at every nonnative number.
Continuity in this text is defined as
Let I be an interval, let f:I→ℝ, and let c be in I. The function f is continuous at c if for each ε>0 there exists a δ>0 such that |f(x)-f(c)|<ε for all x in I that satisfy |x-c|<δ.
The Attempt at a Solution
For x=0 is easy. I've shown that.
Let ε > 0
Find δ > 0 such that |f(x)-f(c)| < ε for all x in I that satisfy |x-c|<δ
I know |x^(1/2)-c^(1/2)| ≤ |x-c| intuitively.
I've tried saying that since |x^(1/2)| ≤ |x| and |-(c^(1/2)| ≤ |-c| => |x^(1/2)| + |c^(1/2)| ≤ |x|+|-c| => |x^(1/2)+c^(1/2)|≤ |x| + |c|
This is where I am stuck. I am not sure how to show this properly.