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I have read that $f(x)=\tan(x)$ is monotonic function.

But in the graph of $f(x)=\tan(x)$ as we move across $\pi/2$ the graph moves from $\infty$ to $-\infty$, i.e graph decreases but the definition for monotonic function says that it should either always increase or decrease.

Wouldn't this make $f(x)=\tan(x)$ non-monotonic? graph of tan(x)

I want to know a reason for this. Please check me if I'm wrong.

Learning
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Mridul Kumar
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2 Answers2

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Your teacher possibly means by “monotonic” that the function is monotonic over any interval completely contained in the function's domain. But I dare say it's not standard terminology.

Of course the tangent function is not monotonic according to the definition

for every $x$ and $y$ in the domain of $f$, if $x<y$ then $f(x)<f(y)$

because $0<3\pi/4$, but $0=\tan0>\tan(3\pi/4)=-1$.

On the other hand, the tangent function is monotonic (increasing) over any interval contained in its domain, because the derivative is $1/\cos^2x$, which is everywhere positive (on the domain), so the mean value theorem applies on every interval inside the domain.

Or your teacher specified that the tangent function is monotonic over $(-\pi/2,\pi/2)$ (so an inverse of the restriction thereon can be defined) and you neglected to note the specification.

egreg
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Though there are confusions about this, as far as I know, we can define monotonicity only on intervals. Looking into Spivak's Calculus:

A function $f$ is [strictly*] increasing on an interval if $f(y)>f(x)$ for all $x$ and $y$ in the interval with $y>x$. (We often say simply that $f$ is increasing, in which case the interval is understood to be the domain of $f$.)

Now, the function $f(x)=\tan x$ is defined on $\mathbb R\backslash \left \{\frac{n\pi}{2}:n\in \mathbb N \right \}$ which is not an interval.

So, monotonicity of $f(x)=\tan x$ on $\mathbb R\backslash \left \{\frac{n\pi}{2}:n\in \mathbb N \right \}$ is not defined.

But, of course the function $g(x)=\tan x$ defined on $\left (\frac{-n\pi}{2},\frac{n\pi}{2}\right)$ is monotonic (increasing in fact).

Sayan Dutta
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    You can define monotonicity on any domain, so long as it is ordered and the codomain is ordered. In particular, if $X$ is ordered and $f: X\to Y$, then $f$ is monotonicly increasing if the following statement holds: $$\forall x_1, x_2\in X: x_1 < x_2 \implies f(x_1) < f(x_2)$$

    there is no requirement that $X$ must be an interval.

     – 5xum Jul 08 '21 at 09:05
  • @5xum I got that idea from Spivak's Calculus, as I added to the post. – Sayan Dutta Jul 08 '21 at 09:09
  • Spivak's Calculus only defines monotonicity on an interval, sure, but that doesn't mean we can't do it on other sets... – 5xum Jul 08 '21 at 09:59