A320 Captain reported an air data malfunction during the initial climb. The Captain considered the weather at the departure airport then made the decision to continue the short leg to the destination airport and landed.

Narrative

During takeoff out of ZZZ climbing though 800 feet we got an ECAM to switch the Capts air data to position 3. We followed the ECAM which restored my air data. This ECAM was followed by more ECAMS letting us know that all the Capts probes had failed. We completed all the ECAMs and checklists. Out of about 10;000 I asked a Captain who was in our jumpseat to back up my FO while I talked to Dispatch and Maintenance Control. I briefed them on the maintenance issue. Dispatch agreed with me that the best course of action was to continue to ZZZ1. ZZZ1 at the time had better weather than ZZZ. I advised dispatch that I felt it would be a good idea to request priority handling into ZZZ1. There were delays due to arrival traffic and I didn't want to get put into holding while having Avionics issues. The weather was clear during cruise and the temperature was well above freezing at ZZZ1 at the time. All of the anti icing failed on the captain's probes as well. Dispatch agreed and we proceeded to ZZZ1. I elected not to brief the flight attendants at this time. If the systems degraded further we would bring them into the loop. After requesting priority handling we were giving priority handling and initially flew an ILS which turned into a visual approach to XXR. Once on the ground we canceled priority handling with ATC and proceeded to the gate.