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Math Help - Contuinity

  1. #1
    MHF Contributor harish21's Avatar
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    Contuinity

    I am confused with some true or false reasoning questions on continuity.

    a. If the function f+g:R -> R is continuous, then the functions f:R -> R and g:R -> R are also continuous.

    b. if the function f^2:R->R is continuous, so is the function f:R->R

    c. If the function f+g:R -> R and g:R -> R are continuous, then f:R -> R is also continuous.

    d. Every function f:N -> R is continuous, where N denotes the set of natural numbers.

    I think that (a) and (c) are true by the definition of continuity. If either f:R -> R or g:R -> R, then according to the definition of continuity, f+g cannot be continuous.

    I am unsure about how (b) and (d) are to be done. Any suggestions?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by harish21 View Post
    I am confused with some true or false reasoning questions on continuity.

    a. If the function f+g:R -> R is continuous, then the functions f:R -> R and g:R -> R are also continuous.

    b. if the function f^2:R->R is continuous, so is the function f:R->R

    c. If the function f+g:R -> R and g:R -> R are continuous, then f:R -> R is also continuous.

    d. Every function f:N -> R is continuous, where N denotes the set of natural numbers.

    I think that (a) and (c) are true by the definition of continuity. If either f:R -> R or g:R -> R, then according to the definition of continuity, f+g cannot be continuous.

    I am unsure about how (b) and (d) are to be done. Any suggestions?
    a) Consider functions on \mathbb{R}:

    f(x)=\begin{cases}0,&x<0\\1,& x\ge 0 \end{cases}

    g(x)=\begin{cases}1,&x<0\\0,& x\ge 0 \end{cases}

    h(x)=f(x)+g(x)=1 and so is continuous on \mathbb{R}

    So we conclude ...

    CB
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  3. #3
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    Quote Originally Posted by harish21 View Post
    I am confused with some true or false reasoning questions on continuity.

    a. If the function f+g:R -> R is continuous, then the functions f:R -> R and g:R -> R are also continuous.

    b. if the function f^2:R->R is continuous, so is the function f:R->R
    Look at f(x)= -1 if x is rational, 1 if x is irrational.

    c. If the function f+g:R -> R and g:R -> R are continuous, then f:R -> R is also continuous.

    d. Every function f:N -> R is continuous, where N denotes the set of natural numbers.
    A function, f(x), is continuous at x= a if and only if f(x_n)\to f(a) for any sequence x_n\to a. If the domain of f is N, the only sequence that converges to a\in N is the constant seqence {a, a, a, a, ...}.

    I think that (a) and (c) are true by the definition of continuity. If either f:R -> R or g:R -> R, then according to the definition of continuity, f+g cannot be continuous.

    I am unsure about how (b) and (d) are to be done. Any suggestions?
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