Hello dear people I noticed this disease on my lemon plant, anyone able to diagnose it? Thanks a bunch. This tree was in my family for more than 40 years. I would appriciate any help.

to the site!
The "disease" is sooty mold which is growing on honeydew deposited by insects, in this case probably scale. You can spray with neem oil or light horticultural oil to help control the insects. You need to spray the whole plant including under the leaves. The scale insects won't kill the plant, but it will look and grow much better.

Thank you @CalPolygardener , I have relayed the information to my father, he is also gratefull. He was aprehensive about the toxicity of neem or horticultural oil. Can he use vinegar solution or something like that?

Vinegar won't work and the oils are safe for edibles.

Indeed after reviewing all the data it would seem that killing the insects is the way to go. Vinegar solution with water would probably just deal with the fungus but not for the root of the problem. Thank you @CPPgardener