Respuesta :
No, It is not possible that f is continuous on [ 4 , 5 ] is non-constant, and takes on only integer values.
Here,
A function f defined on [ 4 , 5 ].
We have to find, is it possible to f is continuous on [ 4 , 5 ], is non-constant, and takes on only integer values.
What is Intermediate value theorem?
Intermediate value theorem states that if “f” be a continuous function over a closed interval [a, b] with its domain having values f(a) and f(b) at the endpoints of the interval, then the function takes any value between the values f(a) and f(b) at a point inside the interval.
Now,
A function f defined on [ 4 , 5 ].
Assume that the function,
f : [ 4 , 5 ] → Z is a non constant continuous function.
Then, f(4) and f(5) both are integers and not equal to each other.
By, Intermediate value theorem;
Function f takes all real numbers between f(4) and f(5) at least once.
These would contain all non integers rational and irrational number between f(4) and f(5), contradicting the assumption that function takes only integer values.
Hence, It is not possible that f is continuous on [ 4 , 5 ] is non-constant, and takes on only integer values.
Learn more about the intermediate theorem visit:
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