How does this result follow for this function?

Question

A function is defined as

\begin{equation}f(x) = \begin{cases} 2, & \text{if } x = 1,2, \ldots,20 \\ 0, & \text{if } x = 21,22, \ldots ,40 \end{cases} \end{equation}

I cannot understand this result: $$f(x) > k \equiv x < c,$$ where $k$ and $c$ are some constants.

Answer 1

Well, the result you propose is incorrect, take k=3 and c=100. So, for x=1, x<100 but f(x)=f(1)=2<3, contradicting the leftward implication, and similarly, for k=0;c=2, x=2 yields an easy contradiction.

Now, showing the commented correction(by Peter Foreman),

For the rightwards implication, fix any x, as f(x)>0, x is an integer strictly between 0 and 21, and thus is definitely less than 21.

For the other side, if x is a natural number less than 21, and in the domain of f(x), then by definition, f(x)=2 > 0.