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Math Help - Analysis Continuous Functions

  1. #1
    Member Jason Bourne's Avatar
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    Analysis Continuous Functions

    Let f,g:R\rightarrow R be continuous functions. Prove directly from the definition of continuity that the function f + 5g is continuous.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Jason Bourne View Post
    Let f,g:R\rightarrow R be continuous functions. Prove directly from the definition of continuity that the function f + 5g is continuous.
    if g is cont., then 5g is cont. so let us prove the case f + h is cont., where h = 5g
    let \epsilon > 0

    f,g continuous, say at x=a \implies \exists \delta _1 , \delta _2 such that if

    i) |x-a| < \delta_1 \implies |f(x) - f(a)| < \frac{\epsilon }{2}

    ii) |x-a| < \delta_2 \implies |h(x) - h(a)| < \frac{\epsilon }{2}


    take \delta = min \{ \delta_1 , \delta_2 \}

    then if |x - a| < \delta,

    |(f+h)(x) - (f+h)(a)| = |f(x) - f(a) + h(x) - h(a)|

     \leq |f(x) - f(a)| + |h(x) - h(a)| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon. QED
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