"Funville," by the way, probably had latitude to make such demands. It was incorporated as a "general law city" about a quarter century ago, though it was considered a "town site" since the late 1800's and likely had its own set of building codes in place by the time you bought. That "general law" distinction gives cities broad powers over local affairs including, most likely, the authority to enforce older codes. As you may know, there are loads of unpermitted additions in Southern California because permitted renovations tend to cause steep property-tax hikes. Hence, many California property owners and their contractors willingly skip permits, which spares you fees and review costs but can, as you now see, create complications later. Such work often goes undetected unless a permitting authority has cause or duty to inspect, which it often does every few years in the case of multiunit rental properties.
Some code authorities will simply agree to sign off on an unpermitted addition on an "as-built" basis if it's safely constructed -- as long as you pony up the fees and, occasionally, fines. Sometimes, modifications are required. However, seldom will code authorities make someone tear down such work, though it does happen! Find out from the building department what steps you need to take. If the city's demands seem excessive, you may need an attorney to run interference. If you seek professional counsel, you might try doing an Internet search for "legalize unpermitted work" where you'll find several real estate and construction attorneys specializing in such problems as well as home-design firms that may offer to walk you through the remediation steps and remap your property. Check references, consumer reviews and Better Business Bureau ratings before proceeding, please.
But don't delay addressing this because that will only compound the problem. In Los Angeles County, for example, rental homes can get thrown into the highly punitive Rent Escrow Account Program, or REAP, if code violations aren't brought into compliance within a set time span as short as 30 days. Under REAP -- or similar programs -- monthly penalties are imposed, tenants can be granted rent reductions and a landlord's eviction rights can become restricted. REAP and similar programs aren't easy to exit either.
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