# Let $f: [0,10) \to [0,10]$ be a continuous function then which is correct

Let $$f: [0,10) \to [0,10]$$ be a continous map then

(a) $$f$$ need not have any fixed point

(b) $$f$$ has atleast $$10$$ fixed point

(c) $$f$$ has atleast $$9$$ fixed point

(d) $$f$$ has atleast one fixed point

Taking counterexample $$f(x) = 1$$; I can easily eliminate option (b) and (c) but I have no idea about first and last options.

Any hints will be helpful

Thank you!

## 2 Answers

Let $$f(x)=10$$ for $$x\in [0, 10)$$.

Take $$f(x)=5+x/2$$. This will eliminate (d). So…

• Thanks for the answer! . Just one question : How did you came up with this counter example so quickly ? It will be helpful for me to know the thought process behind it so that I can use it solve further problems.
– user435638
May 17, 2019 at 12:09
• I drew (in my head) the graph of $x\mapsto x$ (with $x\in[0,10]$) and I thought about what should be the graph of $f(x)$ so that the two graps don't intersect. What I got was a straight line going from $(0,5)$ to $(10,10)$. May 17, 2019 at 12:33