(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

When [itex]|x| < \delta _g[/itex], [itex]|g_r (x) - g_r (0)| < \epsilon '[/itex] ...(1)

When [itex]|x| < 1[/itex], [itex]|g_r (x) - f(x)| < r[/itex] , i.e. [itex]|f(x) - g_r (x)| < r[/itex] ...(2)

[itex]|g_r(0) - f(0)| < r[/itex] ...(3)

Adding (1) and (2) gives [itex]|f(x) - g_r (0)| < \epsilon ' + r[/itex] ...(4)

Adding (3) and (4) gives [itex]|f(x) - f(0)| < \epsilon ' + 2r[/itex]

So when |x| is the [itex] \min\{1, \delta_g\}[/itex] , [itex]|f(x) - f(0)| < \epsilon[/itex]

And by the previous result, we can say that f is continuous at 0.

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Analysis help

**Physics Forums | Science Articles, Homework Help, Discussion**