Can a linear function have zero slope?

Question

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

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. — coffeemath, Dec 13 '21 at 05:11
This function is not linear. Please see my comment below the accepted answer. The only linear function with $0$ slope is $f(x)=0$. — John Douma, Dec 13 '21 at 05:27
Seconding https://math.stackexchange.com/questions/1912970/intutive-difference-between-linear-map-transformation-vs-linear-function. I think that question and answer explains it very clearly. — fleablood, Dec 13 '21 at 05:36

score 1 · Accepted Answer · answered Dec 13 '21 at 05:10

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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.

answered Dec 13 '21 at 05:10

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Thank you. I understood it. – Reza Hosseinzadeh Dec 13 '21 at 05:12
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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. – nicomezi Dec 13 '21 at 05:23
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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. – John Douma Dec 13 '21 at 05:26
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@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. – John Douma Dec 13 '21 at 05:35
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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. – Jacob Manaker Dec 13 '21 at 05:37
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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. – leslie townes Dec 13 '21 at 05:44

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