What could be the reason to cause this discoloration of the leaves? The discolored parts you see here get thicker over the time until the green color disappears. It does not seem that the leaves dry out as the right one on the photo feels quite wet.
This is caused by excessive drying. I have seen this happen when Morning Glory vines found their way into a house and grew up to the window. Even though the temperature didn't go under 65 degrees Fahrenheit (it was summer), every leaf inside the building developed the dead areas on the leaves, but outside they were fine. I thought it was a humidity issue, but discovered a few red spider mites on the plants. The drying effect of the mites plus the low humidity (they usually come together), causes the dry dead patches to form.
Note: The pictures below are of cold damage, not excessive drying.
Cold damage on morning glories comes as a blackening leaf collapse. See the pictures below.
You've said its on a windowsill which has full sun exposure (if there's any sun at all that is). Can I ask if there's a radiator beneath the window, which is currently hot during the day? Do you have double glazing? Do you draw the curtains at night, leaving the plant between the curtains and the window? What size pot is the plant in and how long has it been in that pot, and how tall is the plant currently? It's just this still looks like cold damage to me - if the plant is in a heated room during the day, and is exposed to sun, then it's in very warm conditions indeed. But overnight, when the heating is off, and it's dark, and particularly if its on a windowsill with curtains drawn, that's a very cold place to be indeed, even with double glazed windows, and our temperatures are unusually low in the UK for this time of year.
In my opinion, it could be a too calcareous soil, that causes chlorosis (leaves become white, then dry, then the plant dies)
Try to give to it some Sequestrene iron chelate. Very important to any type of plants, included cacti.
We don't have enough information to provide a definitive diagnosis. Close up pictures would help. Here are some possibilities:
So, check for evidence and put up more photos please!