Does there exist a continuous function that takes each real value exactly 3 times?
An example can be found as the sum of a periodic and a linear function.

## Solution

One possible option is the function \( f(x) = \frac{\mathrm{x}}{3} + \arcsin (\sin \mathrm{x}) \), the graph of which is a "sawtooth" as shown in the image.

![](https://cdn.mathpix.com/cropped/2024_05_06_23a8a0dc60f629d15d84g-34.jpg?height=597&width=1340&top_left_y=90&top_left_x=359)

Another option is the function \( \mathrm{f}(\mathrm{x}) = \operatorname{frac}\{2 \mathrm{x}\}\{3 \pi\} + \sin \mathrm{x} \). This is an example of a "smoothed sawtooth."

## Answer

it exists.