No, Jack/Ernest Worthing has not truly learned "the importance of being earnest" at the end of Oscar Wilde's play The Importance of Being Earnest. The ending of the play is meant to be ironic.

Early in the play, we learn that Jack Worthing has been using the name Ernest while he's in London. He explains to Algernon, "Well, my name is Ernest in town and Jack in the country" (act 1). He claims that he does so because he must guard his reputation due to his responsibility for his ward, Cecily Cardew. He adds, "I have always pretended to have a younger brother of the name of Ernest, who lives in the Albany, and gets into the most dreadful scrapes" (act 1). Algernon understands immediately that what Jack does is similar to his own practice of "Bunburying." Jack emphasizes, however, that he intends to stop using his false persona so that he can settle down and marry Gwendolen Fairfax. Later in act 1, Jack learns that Gwendolen will only marry a man named Ernest, so he plans to be christened by that name as soon as possible. The name Ernest is a pun on the adjective "earnest," which means sincere and genuine. Jack and Algernon are both disingenuous when using the name, so Wilde's pun is also an example of irony.

The men's false "Ernest" personas are exposed at the end of act 2, and the women are horrified to learn they are not named Ernest. However, when the men vow to go through with the christening to change their names, the women seem satisfied. The end of the play, though, exposes the truth of Jack's birth once Lady Bracknell identifies Miss Prism as the woman who used to care for the young Jack and Algy as babies, before she misplaced Jack in a handbag at Victoria Station when he was an infant.

Finally, once Jack knows his true heritage, they all look up his real name, by looking up his father's name in the army registry. It turns out that his name "is Ernest after all" (act 3). He laments that "it is a terrible thing for a man to find out suddenly that all his life he has been speaking nothing but the truth" (act 3). This is meant to be humorous and ironic. Jack has learned nothing; he has basically gotten away with a lifetime of lies. He has been lucky in finding out, quite accidentally, that he truly was who he said he was all along. The ironic twist at the end shows that members of the upper class can live reckless lives with impunity.

In The Importance of Being Earnest, Oscar Wilde's use of epigrams (contradictory statements) is what anchors the comedic aspect of the play. Along with the use of epigrams, Wilde employs irony to drive the situations that twist the plot and that make the the dialogues and dynamics among the characters paradoxical.

The final phrase of the play,

I’ve now realized for the first time in my life the vital Importance of Being Earnest.

is an example of the use of irony and paradox to convey the opposite of what is going on.

Jack is never honest in the play. He never intends to tell anybody his real name until his fake persona is "found out", first by Algernon, and then by Cecily and Gwendolen. Claiming that his name is Ernest for his own convenience, he also lies, to his love interest, Gwendolen, and to Gwendolen's mother, Lady Bracknell.

He is dishonest to Miss Prism, Cecily and Dr. Chasuble telling them that he has a wicked brother in London named Ernest who causes havoc; that, for that reason, he has to go to London to solve Ernest's problems. This is a lie because the reason why Jack goes to London is "to become" Ernest, to run bills in restaurants, and to cause havoc with Algernon. This is why Algernon is shocked in Act I to find out that his friend Ernest's real name is Jack.

Additionally, he is dishonest when he falsely claims that his wicked brother Ernest died from "a chill". This, he does intentionally to end his double life and marry Gwendolen the proper way. Little does he know that Algernon already took the advantage by showing up in Jack's estate pretending to be "the wicked brother Ernest".

All this being said, the play ends with a shocking twist: Jack's real father's name is Ernest and, since Jack is his eldest son, Jack duly gets his father's name. This means that Jack, who lied about being Ernest and about many other things, overrules everything that he did and claims that, after all, he was telling the truth.

When Wilde uses the phrase "the vital importance of being earnest" he does it with two intentions: to point out the irony of the situation, to question the real value of truth versus falsehood. In Wilde's treaty The Decay of Lying, Wilde favors lying as a form of art. Aesthetics favor the fake versus the real; life should imitate art, and not the opposite. In true Wildean fashion, Jack does NOT learn the importance of being earnest. He actually admires the fact that his lying led nowhere, that providence rendered him truthful, and that there will be no consequence as a result of his lies because...they have turned into truths.

By the end of The Importance of Being Earnest by Oscar Wilde, has Jack learned the importance of being Ernest, earnest, or both?

While it is true that by the end of the play Jack has fully taken on the name Ernest, it is harder to say that he really has chosen to be earnest. His lies have all turned out to be true, but he told them believing that they were lies, and it was only through happy coincidence that they turned out to be true. Jack shows an incredible willingness to spin fictions, and the play suggests that whether Jack is seen as earnest or not has more to do with chance than it does with Jack's own values. Truth was less important to him than convenience, and in the end, the truth happened to be convenient. One could argue that he learned the importance of being earnest not as a moral lesson but as a practical one. That is, even if he didn't choose to be honest for honesty's sake, there's a convenience to not telling lies, especially if the truth conveniently happens to be exactly the lie you chose to make up.

By the end of The Importance of Being Earnest by Oscar Wilde, has Jack learned the importance of being Ernest, earnest, or both?

As Gwendolen hints, if Jack has indeed learned anything by the end of the play, he is sure to change. And since Wilde confessed that his purpose in writing The Importance of Being Earnest was to "treat all the trivial things of life seriously, and all the serious things of life with sincere and studied triviality," one must believe that Jack didn't really learn much, and if he did, it wasn't important.

Ignoring all that, one could postulate that Jack learned the importance of being earnest. This follows the spelling in the script. If so, it would mean that by not giving up on his desire to marry Gwendolen and by holding firm against Lady Bracknell, he succeeded in his aim, which was to secure Gwendolen's hand in marriage. He was willing to acquire the name of Ernest one way or another, even by baptism, but because he was so earnest in his devotion to Gwendolen, he was rewarded with the name "naturally."

Could Jack have learned the importance of being Ernest? That is an easier argument to make. Although at the beginning of the play, he doesn't seem much bothered by having "lost" both his parents, when Lady Bracknell advises him to come up with some relations by the end of the season, he perhaps realizes the importance of his missing father. The scrapping he does with Algy is strangely indicative of brotherly "love," and even though he swears he never had a brother, by the end scene he's shouting, "I always said I had a brother!" He seems quite pleased to have a family at last. Learning that his father was named Ernest John brings his identity full circle. He is the man he was as well as the man he pretended to be, so all is well. He has to find satisfaction in that.

One could argue that Jack learned nothing or that he learned everything. Aunt Augusta would probably be pleased with the former since she was against anything that tampers with natural ignorance. But Wilde would perhaps be pleased with the latter, his professed philosophy for the play notwithstanding, since it creates the best wordplay.

By the end of The Importance of Being Earnest by Oscar Wilde, has Jack learned the importance of being Ernest, earnest, or both?

I think Jack has learned both. By the end of the play, Jack understands the importance of being Ernest, the man, because, in learning that his real name actually is Ernest, Jack acquires a family and status that he previously lacked. This family and status pave the way for Jack's marriage to his cousin Gwendolyn. Jack has also come to understand the importance of being earnest, the virtue, because it is ultimately by being himself—instead of pretending to be his brother, Ernest—that puts him in a position that allows him to marry Gwendolyn. Lady Bracknell was unwilling to allow her daughter to marry Mr. Ernest Worthing, but she becomes quite willing to let Gwendolyn marry Mr. Ernest Moncrieff, her sister's son. In the end, being earnest allows Jack to find his way to being Ernest, which enables the play's happy and hilarious end.