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If $|f(x)|$ is continuous at $a$, is $f(x)$ continuous at $a$? I tried doing it using composite functions. If $g(x)= |x|$, then $g\circ f(x)= |f(x)|$. Since $g(x)$ and $g\circ f(x)$ are continuous, $f(x)$ is continuous. I don't know if this is correct. Please help. |
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Let $f(x)=-1$ if $x$ is rational, and let $f(x)=1$ if $x$ is irrational. Or else more modestly let $f(x)=-1$ if $x\lt 17$, and $f(x)=1$ for $x\ge 17$. |
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This is not always true. If $f(x)$ is a piecewise function such that $f(x)=1$ for $x<a$ or $x=a$ and $f(x)=-1$ for $x>a$ Then $|f(x)|$ is continuous at $a$ but $f(x)$ is not. |
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