To "keep something on" and to "leave something on" mean virtually the same thing. Either can refer to an article of clothing that one intends to (or is admonished to) persist in wearing, or to some kind of powered machine (a car, a refrigerator, an electric light, etc.) that one intends to or is requested to permit to remain in the running state.
Both of those can also take "off" instead of "on" as a complement to turn the meaning around into a negative.
Please keep [or leave] the light off. I'm trying to sleep.
To "hang on to" something is a bit different. It can mean one intends to or is requested to not get rid of something.
Hang on to your hopes.
Hang on to that receipt. You'll need that if the IRS audits you.
"Hang on" can also be a request for someone to wait.
Hang on! I'll be there in a minute.
Note that using "off" changes the meaning into something that is not the opposite. It means to literally suspend something or someone from something else.
I wouldn't want to hang off the side of a building until we see world peace.
"Leave off" also can refer to a point where someone stopped doing something:
Where did I leave off in the story I was reading to the kids?
And "keep off" can be a warning against trespassing:
Keep off the grass!
A quote from the MW Dictionary of Synonyms (without examples here):
"Severe is applicable to persons and their looks, acts, thoughts, and utterances or to things (as laws, penalties, judgments, and styles) for which persons are responsible. In all these applications it implies rigorous standards of what is just, right, ethical, beautiful, or acceptable and unsparing or exacting adherence to them; it not only excludes every hint of laxity or indulgence but often suggests a preference for what is hard, plain, or meager (a severe teacher). Very often the word suggests harshness or even cruelty (severe criticism)."
"Harsh suggests a nature that is unfeeling, cruel, and indifferent to the pain it inflicts (a harsh critic) or when applied to things, effectiveness in promoting discomforts or in imposing rigors (a harsh rebuke)."
Compare the following examples (taken from different sources):
The country has come under severe criticism for its human rights record. [strong, neither bad not good]
Many people feel the punishment should have been more severe.
It may seem harsh to punish him, but he has to learn that this kind of behaviour is unacceptable.
He later regretted his harsh words. [negative]
He accused her of being unduly harsh. [negative]
Nothing can justify such harsh treatment of prisoners. [negative]
Best Answer
The two sentances have the same meaning. "On" in the first sentence is just a preposition.Essentially the first sentence is teling you to continue to be engged in challenging yourself, while the sencond sentence is telling you to continue to challenge yourself.