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Mentallic

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I have a constant set of integers [itex][m,n], m>0, m\neq n[/itex] and a variable integer k>0. If we multiply k by successively increasing positive integers t, we will eventually get kt > m. Now, what I want is the output to be kt if [itex]kt \in [m,n][/itex] and n if [itex]kt>n[/itex].

So for example, let's consider m = 45, n = 50 and k = 8.

We get the sequence 8, 16, 24, 32, 40, 48

At this point, we have a value larger or equal to m=45, and 48 < n=50 so the output is 48.

For k=11 we have

11, 22, 33, 44, 55

Since we now have a value larger than m, but the value is also larger than n, the output needs to be n=50.

So finally, my question is, can I create a function that isn't piecewise that defines this process? The piecewise function wasn't such a big deal,

$$

f(k,m,n) =

\begin{cases}

\Big\lceil \frac{m}{k} \Big\rceil k, & \text{if}\hspace{5 mm} \Big\lceil \frac{m}{k} \Big\rceil k < n \\ \\

n, & \text{if}\hspace{5 mm} \Big\lceil \frac{m}{k} \Big\rceil k \geq n

\end{cases}

$$

But I can't figure out how to produce a non-piecewise function.