As a general rule, rivers don't run uphill very often. This may prove to be a slight problem for your scenario, because when talking about mountain ranges, it really IS uphill both ways.
Reality is for those who lack imagination.
Good point with water not running uphill^^ But my idea was more along the line that the rive flows in a valley wich more or less cuts the mountainrange in half.
If the setting is developed enough, you could have a series of locks and canals that serve the same purpose.
Basically a bunch of man-made lakes next to each other, that can be filled or drained of water, going up and down through a pass in the mountain range. You need to go up 50 meters in elevation? Simply raise the water level to that height, to move to the next lake which is drained down to that height. Repeat as necessary.
The Panama Canal, in short.
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Or the river in question may be like the Columbia river in the Northwest. It cuts through not only a large (elevated) basalt province in the Columbia River Plateau, it also cuts clean between the Cascades.
Then you have the Rio Grande river in the American Southwest. It starts west of the Sangre De Cristo Range in Colorado and stays between the Rocky Mountains and other ranges all through Colorado and New Mexico before cutting east through the large canyons of Big Bend Texas before turning into a river that enters the Gulf of Mexico.
Of course both examples have plate tectonics explaining why they happen.
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Maybe the characters just mistake it for running through the mountain range because the valley looks so narrow to the human eye, or is too steep to (safely) live in? Just because they control it doesn't mean they have to live in it.
edited 16th Nov '14 12:36:32 PM by Sharysa
Thank you very much you guys helped me A LOT^^
In my setting a city controlls the only way for ships to reach the sea from the inland and rose to power by taxing all ships that come by. I thought about a mountain range between the inland and the sea, so that it would be too expensiv to transport your wares by land, but with a river wich cuts through said mountain range and the city controlls the valley where the river flows. Now i don´t know anything about geography and if such a thing would actually be possible or if the valley cut by the river would be to wide for one city to control it. Anyone here who could help me with that?