1) Multitasking skills are mostly a myth. Dividing attention is proven to dramatically reduce efficiency. You give me 5 tasks, and I guarantee I will complete them sequentially faster than most people could multitask. The key is triage and focus.
2) Like the others who have commented, I also concur that the 24/7 access of IT is not the norm. Most business would not expect this and frankly, that concept leads to job burnout and training expenses for the company. Any business that needs 24/7 IT would or should have multiple IT personnel working in shifts.
3) As for climbing the corporate ladder, I think this perpetrates the myth of the 'one company career'. Sure, if you are dead set on staying at one company, you will likely hit a wall. However, a career will almost always mean moving upward and onward to other companies as your skill set and experience grows. That should be the exception.
The other points that relate to people skills and temperament are straight on. Giving up is not acceptable. Getting angry is not acceptable. Treating customers as stupid or an inconvenience is not an option.
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