vintwc
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Homework Statement
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.
Homework Equations
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]