In America, we chart the sea changes in our culture through sport. Seminal figures from Jackie to Fernando, Babe to Althea, signal to us that our world is different or, in some cases, make it different all on their own.

“Battle of the Sexes” gives one of those icons her due. It’s not that we’ve forgotten Billie Jean King, a champion whose march to the top heralded the beginning of a golden age of American tennis. But it’s worth remembering, too, her place in feminist history and the blows she struck against sexism and for equal rights and the cartoonish spectacle of a match that made it all possible.

There was a TV movie of almost 20 years ago, “When Billie Beat Bobby,” that got the hilarious hoopla of the 1973 Billie Jean vs. Bobby Riggs match right. What “Little Miss Sunshine” directors Valerie Faris and Jonathan Dayton and “Slumdog Millionaire” screenwriter Simon Beaufoy go for in “Battle of the Sexes” is to connect that match with King’s sexual awakening as a gay woman, and America’s changing attitudes towards that.

And by casting the hard-not-to-be-adorable Steve Carell as the self-promoting “male chauvinist pig” Riggs, they’ve produced a picture that’s alternately giddy and touching, with its heart coming from a budding romance and many of its laughs from the naked, reflexive sexism of the era.

When the most jaw-dropping moment of your movie is special effect footage of the grandiloquent popinjay Howard Cosell draping his patronizing arm over tennis star and commentator for The Big Match Rosie Casals (Natalie Morales) as she delivers her take on the battle, one can’t help but marvel at how apt Madison Avenue’s most famous cigarette slogan of the day still is.

“You’ve come a long way, baby.”

Emma Stone, bronzed and hair-bobbed if not muscular enough to suggest the real King’s athleticism, lets us see the vulnerability beneath the steely competitor. When tennis boss Jack Kramer (Bill Pullman) identifies himself as the true villain of the piece, refusing to pay women anything remotely like what male players earn at American tournaments, Stone’s King doesn’t wilt, weep or swear.

She starts her own tour. Or puts her manager (Sarah Silverman at her feistiest) on it. She talks most of the best women’s players into joining them, and that cigarette company with the sexy slogan, Virginia Slims, sponsors it.

Riggs is a long-faded star who was one of the greatest players of his era, a mind-gaming hustler who won major titles in singles, doubles and mixed doubles in his day. Now, he’s a hilariously unrepentant gambler. He plays cards with his therapist and leads a revolt in his Gambler’s Anonymous meeting.

“Why should we give up the one thing in life we really love?”

His wealthy wife has had enough of this, but in casting Elisabeth Shue in that part, the filmmakers play up the lovable rascal in Riggs. We see him through her amusingly infuriated eyes.

One hustle too many (he wins a Rolls Royce from a rich mark) and she kicks him out. But he sees the Virginia Slims tennis revolt as opportunity. The 55 year-old who plays comic handicapped matches in raingear, wearing scuba fins, or in Little Bo Peep outfits (complete with sheep) will play the top ranked woman player in a hyped big money exhibition match.

Billie Jean isn’t having it. So he plays Australian champ (and new mother) Margaret Court (Jessica McNamee), portrayed here as mercenary and judgmental.

What’s she judging? Billie Jean’s succumbed to the flirtations of an LA hairdresser (the great Andrea Riseborough) and taken her on tour with them. Legendary tennis fashion designer Cuthbert “Ted” Tinling ( Alan Cumming, perfect) may have his Sapphic sister’s back. But Court disapproves, and knows the world would, too.

All of this is folded into events leading up to The Big Match, and while the sexual component does add gravitas and romance to the story, it tends to slacken the pace. The picture plays long, as not all the buildup to the finale has momentum built into it.

Beaufoy makes up for this with some sharp dialogue, most of which is uttered by Cummings, such as the moment when King, upset, realizes this “not a match, a show,” is something she cannot dodge.

“What’s gotten into her?”

“Fate, sweetie, coming at her like a runaway train.”

Stone gives the formidable King the warmth to make her easy to root for. And Carell carries his half of the picture with the pathos of a faded star, willing to vamp away what little reputation he has left for one last stunt in the spotlight.

In the end, King is the embodiment of the change she wants to see in the world and Cosell is the relic of an age so alien that we laugh, in shock, at how commonplace his brand of “little lady” sexism was, way back then.

We’ve all come a long way, babies.

MPAA Rating: PG-13 for some sexual content and partial nudity

Cast: Emma Stone, Steve Carell, Alan Cumming, Andrea Riseborough, Bill Pullman, Sarah Silverman

Credits: Directed by Valerie Faris, Jonathan Dayton, script by Simon Beaufoy. A Fox Searchlight release.

Running time: 2:01