Shutter button stuck

Yes you can. The Olympus SP600UZ can definitely zoom in to the subject not only in still but also while taking videos. The same lever(can be found on the location of the shutter button) that you use for zooming in while taking still pictures, should be the same thing that you access to zoom to your subject video-wise

First try pressing the DISP or DISPLAY button on the back of your camera. If that didn't fix it... A stuck shutter is another common failure mode for digital cameras. The symptoms of a stuck or "sticky" shutter are very similar to CCD image sensor failure. The camera may take black pictures (for shutter stuck closed), or the pictures may be very bright and overexposed, sometimes with lines, especially when taken outdoors (for shutter stuck open). To confirm a stuck shutter, put the camera in any mode other than "Auto", and turn the flash OFF (you don't want to blind yourself for the next step). Next look down the lens and take a picture. You should see a tiny flicker in the center of the lens as the shutter opens and closes. If no movement is seen, then you likely have a stuck shutter. If so, please see this link for further info and a simple fix that may help.

I had the same problem, so tried putting the zoom fully on to take a picture.
It worked.

Is your memory full? Does the camera attampt to focus when you push the button down 1/2 way? If not the mode dial may be faulty (you are not in picture taking mode) or the shutter button may be faulty (no focus attempt) Try to use the timer button to get the shutter to trip.

There are a couple issues related to "slow" that I have with my C-750, and this is how I get around them.

One is the write speed to the xD card. This can prevent you from taking sequential pictures quickly, as the camera displays the current picture it is writing until it finishes. You can make the image of the current picture go away and use your viewfinder again, while the camera is writing the picture, by pressing the shutter button halfway and releasing it. Then you can see your subject in the viewfinder again!

Second is the delay between the time the shutter button is pressed and the time the shutter actually opens (when the camera actually takes the picture). This can be fixed by lining up your shot and pressing the shutter button halfway and holding it for a few seconds. This will give the camera time to adjust and lock focus, shutter speed, etc. There is a small green dot on the display below the battery indicator which will blink a few times when the button is pressed halfway, then remain on steady. When the green dot is steady, the camera is ready. Press the shutter button, and the camera should immediately take the picture.

The "button" is actually just a plastic extension for the actual switch on the printed circuit board. You will have to dismantle the camera to gain access to the board and then use a DMM to test the switch contacts.

Use the shutter button half way until it fucuses and then wait for the moment you want to capture, then press fully.

Changing the firmware won't change anything. The micro switch may be stuck in the on postition. If you are familiar with micro circuits, you could get the switch from Olympus or USCamera.com and replace it. Or, send the camera to Olympus or an Olympus authorized shop.

It sonds like teh shutter is stuck open. If the camera is indera year old it is probably still under warranty. Take it to an authorized repair shop and they will most likely repair it for free.

Try the FOCUS LOCK feature. This feature allows you to focus on a subject roughly the same distance away from you, then re-compose your picture without losing that focus. Open the lens barrier fully. Look through the viewfinder at the AF Target Mark (the 4 lines in the center of the viewfinder). Position the AF Target Mark on a subject roughly the same distance away from you. Press the shutter button halfway until the green lamplights. While keeping the shutter button pressed halfway, re-compose your picture, then press the shutter fully.