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Math Help - showing discontinuity

  1. #1
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    showing discontinuity

    f(x) =

    e^x, x<0
    cos(pi x) , 0 < or equal x < or equal 1
    x^2 - 3/2, x > 1

    I need to prove that f is discontinuous at 1.(by manipulating inequalities, rather than using calculus)

    I have,
    we guess that f is discontinuous at 1, find one {Xn} in A such that

    Xn ---> a but f(Xn) does not ---> f(a)

    I am not sure what sequence to use, can anyone help?
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  2. #2
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    Re: showing discontinuity

    Quote Originally Posted by Arron View Post
    f(x) =

    e^x, x<0
    cos(pi x) , 0 < or equal x < or equal 1
    x^2 - 3/2, x > 1

    I need to prove that f is discontinuous at 1.(by manipulating inequalities, rather than using calculus)

    I have,
    we guess that f is discontinuous at 1, find one {Xn} in A such that

    Xn ---> a but f(Xn) does not ---> f(a)

    I am not sure what sequence to use, can anyone help?
    Well your function is

    \displaystyle f(x) = \begin{cases}e^x\textrm{ if }x < 0 \\ \cos{(\pi x)} \textrm{ if } 0 \leq x \leq 1 \\ x^2 - \frac{3}{2} \textrm{ if }x > 1\end{cases}

    For a function to be continuous at a point, the function needs to approach the same value from the left as it does from the right.

    To the left of \displaystyle x = 1, the function is equal to \displaystyle \cos{(\pi x)}, and so will approach \displaystyle \cos{(\pi \cdot 1)} = -1 as \displaystyle x \to 1 from the left.

    To the right of \displaystyle x = 1, the function is equal to \displaystyle x^2 - \frac{3}{2}, and so will approach \displaystyle 1^2 - \frac{3}{2} = -\frac{1}{2} as \displaystyle x \to 1 from the right.

    Since the function at \displaystyle x = 1 does not approach the same value from the left as it does from the right, the function is discontinuous at \displaystyle x = 1.
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  3. #3
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    Re: showing discontinuity

    Thanks. I also need to prove that F is continuous at 0, do I prove this in the same way?
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  4. #4
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    Re: showing discontinuity

    Quote Originally Posted by Arron View Post
    Thanks. I also need to prove that F is continuous at 0, do I prove this in the same way?
    Yes indeed.
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