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Math Help - continuity - analysis

  1. #1
    Member Jason Bourne's Avatar
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    continuity - analysis

    Let f,g: \Re \rightarrow \Re be continuous functions. Prove directly from the definition of continuity that the function \varphi : \Re \rightarrow \Re defined by

    \varphi (x) = 7f(x) - 3g(x) + 5x

    is continuous
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  2. #2
    MHF Contributor

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    This is real tedious problem. Start with some point a.
    The given \varepsilon  > 0 then for the functions f(x)\,,\,g(x)\,\& \,x
    find three deltas such that:
    \begin{array}{l} 0 < \left| {x - a} \right| < \delta _1 \quad  \Rightarrow \quad \left| {f(x) - f(a)} \right| < \frac{\varepsilon }{{21}} \\ <br />
 0 < \left| {x - a} \right| < \delta _2 \quad  \Rightarrow \quad \left| {g(x) - g(a)} \right| < \frac{\varepsilon }{9} \\ <br />
 0 < \left| {x - a} \right| < \delta _3 \quad  \Rightarrow \quad \left| {x - a} \right| < \frac{\varepsilon }{{15}} \\ <br />
 \end{array}
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