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Math Help - Combinations of Continuous Functions

  1. #1
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    Combinations of Continuous Functions

    (1) Let g(x) = \sqrt[3]{x}.
    (a) Prove that g is continuous at c=0.
    (b) Prove that g is continuous at a point c \not=0. (The identity a^3-b^3=(a-b)(a^2+ab+b^2) will be helpful)

    (2) Using the \epsilon-\delta characterization of continuity (and thus using no previous results about sequences), show that the linear function f(x)= ax+b is continuous at every point of \mathbb{R}.

    (3)
    (a) Using the definition: ( A function f: A \rightarrow \mathbb{R} is continuous at a point c \in A if, for all \epsilon > 0, there exists a \delta > 0 such that whenever |x-c| < \delta (and x \in A) it follows that |f(x)-f(c)| < \epsilon. If f is continuous at every point in the domain A, then we say that f is continuous on A.) show that any function f with domain \mathbb{Z} will necessarily be continuous at every point in its domain.

    (b) Show in general that if c is an isolated point of A \subseteq \mathbb{R}, then f: A\rightarrow R is continuous at c.


    (4) Assume h: \mathbb{R} \rightarrow \mathbb{R} is continuous on \mathbb{R} and let K = {x: h(x) = 0}. Show that K is a closed set.

    If anyone could figure out any of these, I would greatly appreciate it! Thanks!
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  2. #2
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    \forall \epsilon > 0, \exists \delta > 0 such that  |x - c| < \delta \: \: \Rightarrow \: \: |f(x) - f(c)| < \epsilon

    1.(a) c = 0:
    |f(x) - f(c)| = |\sqrt[3]{x} - \sqrt[3]{0}| < \epsilon
    |\sqrt[3]{x}| < \epsilon
    |x| < \epsilon^{3}

    We know that |x - c| = |x - 0| = |x| < delta and we know that |x| < epsilon cubed. What should we choose delta to be?

    (b) c not 0:
    |f(x) - f(c)| = |\sqrt[3]{x} - \sqrt[3]{c}| < \epsilon
    \iff |x^{\frac{1}{3}}-c^{\frac{1}{3}}| \cdot \left|\frac{ x^{\frac{1}{9}} + x^{\frac{1}{9}}c^{\frac{1}{9}} + c^{\frac{1}{9}} }{x^{\frac{1}{9}} + x^{\frac{1}{9}}c^{\frac{1}{9}} + c^{\frac{1}{9}} }\right| < \epsilon
    (Think of x - c as a difference of cubes. So, the x^{\frac{1}{3}} - c^{\frac{1}{3}} corresponds to the (a-b) part of a^{3} - b^{3} = (a-b)(a^2 + ab +b^2))

    \iff \frac{|x-c|}{|x^{\frac{1}{9}} + x^{\frac{1}{9}}c^{\frac{1}{9}} + c^{\frac{1}{9}}|} < \epsilon

    etc.

    2. |f(x) - f(c)| < \epsilon
    |ax + b - (ac + b)| < \epsilon
    |ax - ac| < \epsilon
    |a| \: |x - c| < \epsilon
    |x-c| < \frac{\epsilon}{|a|}

    and the conclusion follows.
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    Quote Originally Posted by shadow_2145 View Post
    (4) Assume h: \mathbb{R} \rightarrow \mathbb{R} is continuous on \mathbb{R} and let K = \{x: h(x) = 0\}. Show that K is a closed set.
    Hint: If A is a closed set then h^{-1}(A) would be a closed set.
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