Air Carrier Flight Crew flying EMB-145 aircraft reported Anti-Ice System failure.

Narrative

Climbing out from ZZZ we experienced an Engine 2 Anti Ice failure. We experienced icing in the climb out and requested higher to exit the icing when presented with the E2 Anti Ice fail. The first flight of the day checks showed no signs of the icing systems not working properly prior to takeoff. The ice system failed nearly immediately after entering icing conditions. We were cleared to climb and ATC requested a top for our climb; this was an ongoing request in search for skies free of icing. As we continued with the flight; we worked with ATC for higher to remain clear of icing conditions. I was the Pilot Flying and assigned the First Officer to troubleshoot. We ran the Quick Reference Handbook and verified that the E2 Anti Ice system had failed. On confirmation of the failure after the completion of troubleshooting via the Quick Reference Handbook; we confirmed with ATC our requirements and worked with dispatch. We chose to continue to ZZZ1 due to better weather and the ability to remain clear of icing conditions. This was first the safest option; following that being the safest option it did not create any undue burden to the passengers or operation. ATC worked with us to remain clear of clouds/icing and worked to give us a block altitude for cruise and pilot's discretion for descents. We were able to remain clear of icing conditions as we continued to our destination of ZZZ1. ATC did an amazing job working with us. We had minor altitude deviations remaining clear of clouds. ATC did assign a block altitude as well to assist with keeping us clear of all icing conditions. ATC had no issue with our altitude variation with our requirement(s) to remain clear of icing for the icing system failure. I am unsure of the greatest difference in altitude vs assignment but none were gross or requiring comment from ATC. Potentially request and be granted block altitudes earlier in the process of the [priority]. Likelihood is not strong but worth considering if in the same event in the future.