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Math Help - Help in Limits of Continuity...

  1. #1
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    Help in Limits of Continuity...

    I need help in proving those problems.
    Please help..

    Q1) Suppose f: [a,b] ->R is continuous and that f([a,b]) is subset of Q. Prove that f is constant on [a,b] using Intermediate value Theorem.

    Q2) Prove that 2^x = 3x for some x E (o,1).

    Q3) Let f(x) = (x^2 - 4x - 5) / (x-5) for x not equal to 5. Can f(5) be defined so as to make f continuous at 5.


    Thanks!
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by Vedicmaths View Post
    I need help in proving those problems.
    Please help..

    Q1) Suppose f: [a,b] ->R is continuous and that f([a,b]) is subset of Q. Prove that f is constant on [a,b] using Intermediate value Theorem.

    Q2) Prove that 2^x = 3x for some x E (o,1).

    Q3) Let f(x) = (x^2 - 4x - 5) / (x-5) for x not equal to 5. Can f(5) be defined so as to make f continuous at 5.


    Thanks!
    Still thinking about 1)

    2) define g(x)=2^x-3x So g is continous on [0,1] and
    g(0)=1 \\\ g(1)=-1 so by the intermediate value theorem There exisits a c \in (0,1) such that g(c)=0

    g(c)=0 \iff 0=2^c-3c \iff 3c=2^c

    3) f(x)=\frac{(x-5)(x+1)}{(x-5)} = x+1 \mbox{ if } x \ne 5

    so if we define f(5)=6 We get what we want
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  3. #3
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    For #1, note that between any two numbers there is an irrational number. Being continuous the function has the intermediate value property.
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  4. #4
    MHF Contributor red_dog's Avatar
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    Suppose f is not constant. Then \exists x_1,x_2\in[a,b], \ x_1\neq x_2 such as f(x_1)=\alpha, \ f(x_2)=\beta, \ \alpha\neq\beta.
    Let c\in(\alpha, \beta)-\mathbf{Q}.
    f continuous \Rightarrow \exists x_3\in(x_1,x_2) such as f(x_3)=c.
    But, f(x)\in\mathbf{Q} \ \forall x\in[a,b], contradiction.
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