One - half of a convex …

Yes it will. Even if half of the convex lens is covered with black paper, the lens will produce a complete image. however the intensity of the image may be less. It can be observed experimentally by using a lighted candle and a convex lens.

Experiment - 

1. Take a lighted candle and place it in front of a convex lens mounted on a plane surface.

2. Move the candle along the axis of the plane to obtain the full image on the screen. Once the full image is observed, mark the position of the candle without moving it.

3. Now cover the lower exact half of the lens with a black opaque paper. Do not change the position of the candle.

4. At this point, you will observe a full image of the candle, but you will find that the intensity is reduced. This is because the covered part of the lens does not allow light to pass through it. So the amount of light reaching the screen is reduced.

Diagram -