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Can we say function $$y=f(x)=10$$ is a linear function? Can a linear function have zero slope?

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    $\begingroup$ Well it is a constant function. There is a definition of "linear function" which would only include $f(x)=cx$ but not with a nonzero constant. $\endgroup$
    – coffeemath
    Commented Dec 13, 2021 at 5:11
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    $\begingroup$ math.stackexchange.com/questions/1912970 $\endgroup$
    – 311411
    Commented Dec 13, 2021 at 5:16
  • $\begingroup$ Thank both of you $\endgroup$ Commented Dec 13, 2021 at 5:24
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    $\begingroup$ Seconding math.stackexchange.com/questions/1912970/…. I think that question and answer explains it very clearly. $\endgroup$
    – fleablood
    Commented Dec 13, 2021 at 5:36
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    $\begingroup$ It is affine. Had to get my spake in. $\endgroup$
    – copper.hat
    Commented Dec 13, 2021 at 5:51

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By definition, a linear function is a polynomial of degree one or less, including the zero polynomial. $f(x)=10$ has degree $0$, so it is linear.

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    $\begingroup$ I have to point that this definition may vary in some languages. In french, a linear function is of the form $f(x)=ax$ and $g(x)=ax+b$ is not linear, but affine. $\endgroup$
    – nicomezi
    Commented Dec 13, 2021 at 5:23
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    $\begingroup$ A linear function $f$ is one such that $f(x+y)=f(x)+f(y)$ and $f(ax)=af(x)$. $f(x)=10$ is not linear. $\endgroup$
    – John Douma
    Commented Dec 13, 2021 at 5:26
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    $\begingroup$ @SeanXie Most of us are mislead into believing that in our early education. For example, $f(x)=x$ is linear but $f(x)=x+1$ is not. This is because $f(0)=f(0+0)=f(0)+f(0)\implies f(0)=0$ for a linear function $f$. So, for example, $y=mx$ is a linear function but $y=mx+b$ where $b\ne 0$ is not. $\endgroup$
    – John Douma
    Commented Dec 13, 2021 at 5:35
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    $\begingroup$ It is conventional in US middle-school education to call affine functions linear. I suspect that's the level of the OP, which makes this a perfectly fine answer, unworthy of deletion. $\endgroup$ Commented Dec 13, 2021 at 5:37
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    $\begingroup$ As Jacob notes, there are multiple definitions of "linear" in use, and many U.S. K-12 and college curricula (pre-linear algebra) would say that a polynomial function of degree at most 1 is a linear function. Having different uses of the same word in common circulation is not quite the same thing as people being "misled" into error. $\endgroup$ Commented Dec 13, 2021 at 5:44

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