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  1. #1
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    analytic function

    let g:[0,1] \rightarrow R be a continuous function. Let \epsilon >0. Prove that there is a real analytic function h: [0,1] \rightarrow R such that |g(x)-h(x)| < \epsilon for all x \in [0,1].
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  2. #2
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    Quote Originally Posted by Archi View Post
    let g:[0,1] \rightarrow R be a continuous function. Let \epsilon >0. Prove that there is a real analytic function h: [0,1] \rightarrow R such that |g(x)-h(x)| < \epsilon for all x \in [0,1].
    You can even find a polynomial that will do this (that is the Weierstrass approximation theorem). A concrete way to construct a polynomial that approximates a given continuous function to any desired degree of accuracy is to use Bernstein polynomials.
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  3. #3
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    can i just say it is proven by weierstrass approximation theorem? or do i have to actually show the proof ? i am not sure how to even start. whould u guide me?
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