Varify if it is true or False and prove it If f is defined o

Varify if it is true or False and prove it.

If f is defined on R and f(K) is compact whenever K is compact, then f is continuous on R.

Solution

The statement is FALSE

Since we can define a function f(x) which will be compact for every K but still the function won\'t be continuous

Proof Using Counter Example

Let we define a function f(x) such that f(x) = -1 for all negative numbers and +1 for all positive numbers

f(x) = -1, -inf < x < 0

1, 0 < x < inf

Hence the function is discontinuous at x=0, but the function is compact, hence the given statement is FALSE