Puzzle: Finding a Bounded Continuous Function

This is a simple challenge. Leave your thoughts in comments.

Puzzle 1: Find a continuous f:\mathbb{R}\to\mathbb{R} with its range contained in (0, 1).

Hint/Puzzle 2: Find the equation of the function below.


Clue 1: Piecewise functions may also be continuous.
Clue 2: Try manipulating e^x

You can use Desmos or Geogebra to plot the graphs and see.

In Geogebra, to plot a piecewise function, use

 Function( <expression in x>,<lower bound>,<upper bound>)

For example,


(For Teachers) Aim of the Activity: 1. Familiarising students with visualising concepts/using software like GeoGebra.
2. Training students to generate examples, given conditions.

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