Falcon Captain reported while climbing on departure experiencing engine failure that was the result of fuel exhaustion related to a fuel gauge issue. The flight continued to its destination airport.

Narrative

Type of Event: Engine Failure (Fuel Exhaustion-No.3 Engine)When and Where: XA:10. Location: ZZZ. Phase of Flight: Initial Climb (8000ft)Aircraft and Role: Dassault Falcon 900B (Part 91) Role: Captain (PIC (pilot in command); non-flying pilot)During initial climb out of ZZZ Airport at approximately 8;000 feet; the flying pilot noticed a reduced rate of climb and lower than expected airspeed and thrust performance. Upon scanning engine instruments; we both observed that the No. 3 (right) Engine had failed completely. ATC communications were maintained with ZZZ TRACON followed by ZZZ Center.Subsequent review indicated that the failure was due to fuel exhaustion in the No.3 Tank. The aircraft had a known discrepancy with the fuel gauge only; not the fuel system. It had been deferred as 'known issue' by operator after multiple attempts of resolving issue by maintenance facilities. Ultimately; it appeared that the problem extended beyond the gauge itself. The No.3 Tank does not uplift fuel from single point panel when set to 'partial'. Generally speaking; this is how I've fueled this type of aircraft for the past decade (complacency). You simple set the panel and walk away and verify with fuel gauges. However; when the fuel panel works correctly... (when panel set to partial; we simply set the amount of LBS in fuel required for mission. When set in Full we request number of gallons required. system does not stop automatically). Unfortunately; this (fueling ;not gauge) issue was was not communicated to the flight crew. The crew assumption was that the tank quantities and usage would be reflected similarly across all tanks such as practiced in simulator when having an inflight gauge issue; i.e.; oil PSI; N1 etc.; (with close monitoring of other connected systems). In conclusion; we had an inoperative fuel gauge with no indication if the tank had fuel. Contributing Factors:Known but uncorrected aircraft discrepancy(fuel gauge problem);Incomplete information provided by the operator regarding No.3 tank fuel characteristics while uplifting fuel;Assumption regarding fuel tank status and system behavior; Complacency and over-reliance on operators assurances;Judgment lapse in trust and accepting the aircraft with know fuel gauge indication issue due to complacency; Confirmation Bias with assumption of all three fuel tanks being equal. Assuming this; we had a known amount of fuel on board (math in gallons uplifted worked out with this assumption).Corrective Actions:The Crew (PIC; SIC (second in command)) secured the failed engine via Abnormal Procedures Checklist and continued the flight without further incident. Moving forward; I will no longer accept an aircraft with unresolved or inadequately explained fuel system discrepancies per MEL. More rigorous verification of fuel feed and system operations will be conducted prior to departure; and assumptions will not replace direct verification. I will self evaluate and confirm there is no complacency.