Dusty Telescope

Relevant wiki: Lenses

Although half of the light reaching the telescope lens is blocked, light from all parts of the moon will still reach the clear part of the lens and will form the image seen by the observer. So the full moon will still be visible, although the image will be less intense than if the lens was clean.

Relevant wiki: Lenses

A telescope uses converging lens, which converges infinite light rays into a point to form an image. In this case, the object comes from infinity, meaning that the light rays coming from the object will be almost parallel (in this case, they are so almost parallel we assume them to be parallel).

Note that without the blockage of dust on the lower half of the telescope, the image will be formed by the convergence of infinite light rays onto a point to form an image. Now add the blockage. We can see that there will still be half the number of the original light rays converging on that same point , forming the same image. Hence, an image of the full moon will still be observed , but it will only be half as bright.

Relevant wiki: Lenses

It doesn't matter how much of the lens is covered. If we look at anything with half the telescope lens covered, we can still see the whole thing, we just can't see the other half - covered by dust.

For a visual example, picture yourself looking through a clean telescope at the moon (which is quite far away). Then imagine the lens half-covered with dust and look through it, you can still see the moon with the clean half. Next, imagine the telescope is built like that - it only has half a lens - then look through it, you can still see the moon.

Light from all points on the moon will pass through any given part of the objective lens before being focussed to the viewer's eye. Obscuring part of the lens will make the entire image dimmer rather than hiding part of the image.

if $\frac{1}{2}$ or more than $\frac{1}{2}$ portion of a lens is OK,then we can still get a full view of something.

example-we can see the full moon with both eyes or with one eye.

This is the same kind of conceptual question I would ask a class of optics students. The action of the lens is to focus light so that you can see it in your eye. It does not matter which part of the lens light from the moon falls on, since each point on the surface of the lens is responsible for getting the rays to your eye. Thus, you will see the full moon, but at about half the brightness .

It's also important to note that any object that is out of focus will not have symmetric blurring. The "bokeh" will be distorted and look like half circles.

The image resolution might suffer a bit along one direction due to the input aperture being only half a circle. Thus, any diffraction limited telescope will not produce the same resolution long the direction that is half covered.

A point for further reading is there are some powerful computational optics techniques that exploit the ideas introduced here to do some powerful imaging: See this link for more information

The moon will look essentially the same as through a telescope of smaller aperture. The shape of the opening has no effect. Reflecting telescopes have a mirror blocking the light path and they work just fine.

If we have covered half of the lens then the brightness of the moon will decrease not the size of the moon

We can see full picture if only any part of the lense is clean.