# Homework Help: Sketching a graph that meets given condition

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1. Apr 12, 2016

### The Subject

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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?

2. Apr 12, 2016

### SammyS

Staff Emeritus
Is it continuous on (0,1) ?

3. Apr 12, 2016

### The Subject

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

4. Apr 13, 2016

### a_Vatar

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

5. Apr 13, 2016

### Ray Vickson

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.