Crowds gathered in towns and cities following the murder of the 23-year-old teacher.
Vigils have taken place across the island of Ireland to remember murdered primary school teacher Ashling Murphy.
Ms Murphy’s family attended a candlelit vigil near the murder scene on Friday evening.
At the event, her father Ray Murphy paid a poignant tribute to the talented young musician by performing her favourite song on the banjo.
He broke down in tears while playing the final chords of When You Were Sweet Sixteen.
Mr Murphy, along with his wife Kathleen and daughter Amy walked on the opposite side of the canal to where his daughter was assaulted and died on Wednesday.
Irish police are continuing to hunt for the killer of Ms Murphy, who was found dead after going for a run on the banks of the Grand Canal in Tullamore, Co Offaly.
The Garda said it had made “significant progress” in its investigation amid reports detectives had identified a person of interest. Gardai said they were not releasing details for operational reasons.
Gardai also said the Murphy family were “appreciative and overwhelmed by the national outpouring of support shown to them”.
Taoiseach Micheal Martin was among those who stood in silence outside the Dail parliament in Dublin on Friday, while Northern Ireland’s deputy First Minister Michelle O’Neill joined the crowd at City Hall in Belfast.
The events were held as the President of Ireland Michael D Higgins paid his own tribute to Ms Murphy and called for society to reflect on what is needed to eliminate violence against women.
“The outpouring of grief at the death of Ashling shows how we have all been very touched, and it is so exemplary for young and old to read of all Ashling’s accomplishments during her short but brilliant and generous life,” he said.
More vigils and memorial events will be held in the days to come.
In Tullamore, shops, businesses and cafes closed early on Friday, as the midlands town came largely to a halt as thousands attended the vigil in a local park.
There were many tears as people of all ages came to mourn and pay their respects to the teacher.
Friends of Ms Murphy were among the traditional Irish musicians who played at the vigil.
Local priest Father Joe Gallagher addressed the crowd before calling for a minute’s silence. The gathering was attended by all the main religious groups from the area.
“We remember her heartbroken family, her colleagues in work, in music, in sport, in friendship and her young pupils in first class who loved their teacher,” said Fr Gallagher.
“This is a time of grief beyond words. We need to be together. We need to support one another in this dark time.
“We stand together, united with groups all over our country, and indeed beyond, united with women who fear and know the trauma of violence. United in grief, in anger, in shock.
“In this dark evening we want to hold a light in our hands, to stand together in solidarity with one another to share our tears and deep grief. Time to pray, to reflect, to listen, to be together.”
Attracta Brady, who was Ms Murphy’s first fiddle teacher, played alongside other musicians at the vigil in Tullamore.
“She was the most beautiful girl inside and out,” she said.
“She was a parent’s dream. She was everything you’d want in a daughter. She had integrity, she was honest, she was trustworthy. She was quirky and a little bit cheeky sometimes, with the loveliest smile, and she’d get away with it because she had this beautiful twinkly smile.
“She was never in bad humour, she was always smiling and she absolutely loved her fiddle. Her parents only told me yesterday that she never had to be told to practise. She was bright and energetic and everybody loved her.”
Earlier, Taoiseach Mr Martin said that the murder had “united the nation in solidarity and revulsion”.
“No stone will be left unturned in terms of bringing this investigation to a completion and to bring the person responsible for this to justice,” he said.
Politicians have promised that all resources necessary will be provided to the gardai to find the killer.
The death of Ms Murphy has sparked fresh debate about the safety of women in Ireland, with many asking how such an attack could happen in broad daylight.
“We, as a society, need to face up to this. There is an epidemic of violence against women. It’s been going on for millennia, quite frankly,” Tanaiste Leo Varadkar said on Friday.
“Men and boys I think, in particular, have a responsibility to start to have that conversation among ourselves about the kind of factors, the kind of attitudes, that give rise to feelings that engender men to commit acts of violence against women.”
Gardai continue to appeal for witnesses and ask anyone with information about a bicycle – a Falcon Storm mountain bike with straight handlebars and distinctive yellow/green front forks – to come forward.
In a statement on Friday evening, gardai said: “Significant progress has been made in the investigation to date.“An Garda Siochana is not confirming any specific details for operational reasons.
“An Garda Siochana continue to support Ashling’s family at this time. The Murphy family are appreciative and overwhelmed by the national outpouring of support shown to them.
“The Murphy family have requested that they now need privacy, space and time to process Ashling’s death.”
On Thursday, officers released a man they had been questioning, saying he was “no longer a suspect”.
The man’s solicitor told the PA news agency that he has had his “life ruined”.
Donal Farrelly, who represented the man during his two days of questioning, condemned those who had tried to identify him on social media.
The town of Tullamore has been left reeling from the death of Ms Murphy, who was described as a “special girl” and a “little angel” by her family.
In an interview with the Irish Independent newspaper, her father Raymond said: “She was a great worker, with great drive. A marvellous musician.
“She crammed so much into her short life.”
Those who knew her described her as a gifted musician who was loved by her pupils.
It is believed about 50 officers are working on the investigation and a post-mortem examination has been completed.
The route along the Grand Canal is often busy and is a popular spot for walkers and joggers.
Floral tributes were left outside the gates of Durrow National School, where Ms Murphy taught, and on Friday the school issued a fresh tribute to her.
In a statement posted on Twitter, the school said it was “utterly devastated by the passing of our dear colleague and friend”.
“Ashling was a very professional and talented young teacher. We are deeply saddened by her tragic loss. Our thoughts are also with her beloved family at this sad time.”