TIFR-2012-Maths-D: 31

Question

True/False Question:

$f : \left [ 0,\infty \right ]\rightarrow \left [ 0,\infty \right ]$ is continuous and bounded then $f$ has a fixed point.

Answer 1

Bounded Function: A function is said to be bounded if it is upper as well as lower bounded.

Fixed point : x is called fixed point of a function f if, f(x)=x  i.e. fixed point is an element in domain of f which is mapped to itself.

Since f(x)=x, so to find fixed point x, you need to find solution of f(x)-x=0. Graphically, fixed point is the point of intersection of graphs of y=f(x) and y=x.

Because f is bounded and continuous in [0,∞], you can clearly see that graph of y=f(x) will surely intersect y=x for some x in [0,∞]. Note that f must be bounded as well as continuous to guarantee that it intersects with y=x and hence has a fixed point.