# Sketching a graph that meets given condition

## Homework Statement

Sketch the graph of a function f that is defined on [0,1] and meets the given conditions (if possible)

- f is continuous on (0,1), takes on only two distinct values.

## The Attempt at a Solution

https://scontent-lga3-1.xx.fbcdn.net/v/t34.0-12/13020084_1115236431831934_1952173744_n.jpg?oh=36c9b39d36f9d87dbfe189161ecdf210&oe=570FDD68

The solutions manual said it is impossible.

What is wrong with this function?

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

Sketch the graph of a function f that is defined on [0,1] and meets the given conditions (if possible)

- f is continuous on (0,1), takes on only two distinct values.

## The Attempt at a Solution

[ IMG]https://scontent-lga3-1.xx.fbcdn.ne...=36c9b39d36f9d87dbfe189161ecdf210&oe=570FDD68[/PLAIN]

The solutions manual said it is impossible.

What is wrong with this function?
Is it continuous on (0,1) ?

So, intuitively no, since "i lifted my pen while drawing this function".

(i) the function f is defined at a
Yes

(ii) the limit of f as x approaches a from the right-hand and left-hand limits exist and are equal
If a is the point that jumps, is the lim x-> a = 1 (correct?)

(iii) the limit of f as x approaches a is equal to f(a).
Iim x-> a = 1 does not equal f(a)=2, no

I see

A continuous function on an interval(in R), should possess an intermediate value property. That's why it's impossible

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

Sketch the graph of a function f that is defined on [0,1] and meets the given conditions (if possible)

- f is continuous on (0,1), takes on only two distinct values.

The solutions manual said it is impossible.

Whether or not it is possible depends on exactly how the problem's wording is interpreted.
Interpretation (1): f is defined on [0,1] and takes two values on that set. It is continuous on (0,1).
Interpretation (2): f is defined on [0,1]. It is continuous on (0,1) and takes two values on that set.

Interpretation (1) is possible, but Interpretation (2) is impossible, for reasons explained already by others.