- #1
sbashrawi
- 55
- 0
Homework Statement
Let f be a nonconstant function. Prove that f has at most countably many zeros
sbashrawi said:You mean that we can prove it by contradicion.
Let S be the set of zeros of f and suppose that it is uncountable.
then we can get a sequence of these zeros convergent to a point in z.
This gives that the zeros are not isolated.
but, if this is what you mean, how can we be sure that we have a convergent series in S.
sbashrawi said:suppose that f doesn't have caountably many zeros.
then it has infinitely many zeros in one of this partition.
Infinitely many zeros in a bounded set implies an accumulation point of zeroes, and an accumulation point of zeros for an analytic function implies that that function is zero everywhere.
contradiction
So f has at most countably many zeros.
sbashrawi said:I think this comes from the infinitely many zeros ( uncountable assumption)
An analytic function is a mathematical function that is defined and continuous over a certain domain. It can also be differentiated an infinite number of times within that domain.
Zeros, also known as roots, of an analytic function are the values of the independent variable that make the function equal to zero. In other words, they are the values that satisfy the equation f(x) = 0.
One way to find the zeros of an analytic function is by solving the equation f(x) = 0 using algebraic methods. Another method is by graphing the function and identifying the x-intercepts, which represent the zeros.
Yes, an analytic function can have complex zeros. This means that the zeros have both a real and imaginary component. Complex zeros are important in areas of mathematics such as complex analysis and number theory.
The zeros of an analytic function can be represented on the graph as the x-intercepts. This means that the graph will intersect the x-axis at the points where the function equals zero. Additionally, the number of zeros of a function can also provide information about the behavior and shape of the graph.