Continuity of $f(x)$ at 0 Using $g_r(x)$

[vintwc]

Homework Statement

If f:\mathbb{R} \to \mathbb{R} is such that for all r>0 there exists a continuous function g_r \mathbb{R} \to \mathbb{R} such that |g_r (x) - f(x)| < r for |x| < 1 then f is continuous at 0.

Homework Equations

The Attempt at a Solution

When |x| < \delta _g, |g_r (x) - g_r (0)| < \epsilon ' ...(1)
When |x| < 1, |g_r (x) - f(x)| < r , i.e. |f(x) - g_r (x)| < r ...(2)
|g_r(0) - f(0)| < r ...(3)

Adding (1) and (2) gives |f(x) - g_r (0)| < \epsilon ' + r ...(4)
Adding (3) and (4) gives |f(x) - f(0)| < \epsilon ' + 2r
So when |x| is the \min\{1, \delta_g\} , |f(x) - f(0)| < \epsilon
And by the previous result, we can say that f is continuous at 0.

Not sure whether its right for me to take the \delta = \min\{1, \delta_g\}

 

[lanedance]

I think it is ok, though it doesn't make much sense to have delta greater than 1 anyway...