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Thread: True/False - continuous function

  1. #1
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    True/False - continuous function

    The following are true or false variances of determining whether a function is continuous.

    1. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(K) is bounded, then f is continuous.
    False. A bounded step function such as the characteristic function would serve as a counterexample.

    2. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(K) is sequentially compact whenever K is sequentially, then f is continuous.

    False. The characteristic function serves as a counterexample
    $\displaystyle f(x)=
    \begin{cases}
    1 & \text{if $x \in K$}, \\
    0 & \text{if $x \notin K$}.
    \end{cases} $ since it is bounded and closed, but it is not continuous.

    (I know the converse is true: if f is continuous and K is sequentially compact, then f(K) is sequentially compact. )

    3. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(A) is closed whenever A is closed, then f is continuous.
    False. Consider the function
    $\displaystyle f(x)=
    \begin{cases}
    tanx & \text{if $ - \frac{\pi}{2} < x < \frac{\pi}{2}$}, \\
    0 & \text{if $otherwise$}.
    \end{cases} $
    It is closed on the real line but it is not continuous. Is there a better counterexample for this?

    4. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(A) is open whenever A is open, then f is continuous.
    False. I was thinking f(x) = 1/x. The range is $\displaystyle (- \infty, 0) \cup (0, \infty) $, which is open, but f is not continuous.

    Thank you.
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  2. #2
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    Quote Originally Posted by Paperwings View Post
    1. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(K) is bounded, then f is continuous.
    False. A bounded step function such as the characteristic function would serve as a counterexample.
    I would use the Dirichlet function. It is defined to be $\displaystyle 1$ for rationals and $\displaystyle 0$ for irrationals. It is nowhere continous.

    2. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(K) is sequentially compact whenever K is sequentially, then f is continuous.
    Again use Dirichlet.

    3. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(A) is closed whenever A is closed, then f is continuous.
    Again use Dirichlet.

    4. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(A) is open whenever A is open, then f is continuous.
    Define $\displaystyle f(x) = 1/x$ and $\displaystyle f(0)=0$.
    Then $\displaystyle f(\mathbb{R}) = \mathbb{R}$.
    And it is clearly not continous.
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