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I'm inspecting a function as in the title, and tried to plot it and then compare my result with graphing tools which lead me to confusion.

Is it true that the plot of $f(x) = 1 + \sqrt{\log_{10}\cos(2\pi x)}$ is just a set of points $(x, y) = (n, 1); \; n \in \mathbb Z$

Inspecting the domain one may see that $$ \log_{10}\cos(2\pi x) \ge 0 \iff \cos(2\pi x) \ge 1 $$

But $\cos x \in [-1, 1]$ therefore $\cos(2\pi x)$ must be equal to $1$ in order to satisfy the above, and that is only possible in $x \in \mathbb Z$.

The reason I'm asking is because neither Desmos nor W|A is plotting it the way i expected.

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  • $\begingroup$ Perhaps they are plotting an imaginary component, $i\sqrt{-\log_{10}\cos(2\pi x)}$ $\endgroup$
    – Empy2
    Commented Jul 25, 2018 at 12:17

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You are right: the domain of $f$ is the set $ \mathbb Z$ and the graph of $f$ is given by

$ \{(x,f(x)): x \in \mathbb Z\}=\{(n,1): n \in \mathbb Z\}$.

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