1. One more: about income taxes

In the country of Bxwy, the tax rate on income below $10,000 is 0%, the tax rate on income between$10,000 and $30,000 is 4%, the tax rate on income between$30,000 and$75,000 is 12%, the tax rate on income between$75,000 and $120,000 is 19%, and the taxrate on income above$120,000 is 25%. The tax law also states that the total tax owed mustbe a continuous function of income.
Describe the (piecewise-de ned) function
f(x) as clearly as possible by formulas. Here

x
is the income of a resident of Bxwy, and f(x) is the tax owed on that income. Graph

y = f(x) for x between 0 and 150,000.

I started with f(x)=0 for x<10000, .04x for 10,000<=x<=30000, .12x for 30000<=x<=75,000 etc.

Unfortunately this is a step function....how do I get this to be continuous? i.e. lim x->30000+=30,000*.12, but lim x->30000-=30,000*.04. Do I need to make it f(x)=.04x+a for 10,000<=x<=30000 and =.12x+b for 30,000<=x<=75,000 and so on and solve for the constants? That doesn't seem like it would get the correct result though as it wouldn't be 12%, it would be 12% and then some...essentially, how do I properly make this continuous? Thank you.

2. My guess concerning 12% rate between $30,000$75,000 is that one pays 12% only on the portion of the income that exceeds $30,000. Similarly, one pays 4% on the portion of the income between$10,000 and $30,000. So, if the income is$20,000, the tax is 4% * ($20,000 -$10,000). If the income is $31,000, then the tax is 4% * ($30,000 - $10,000) + 12% * ($31,000 - $30,000). Hint: the first term in this sum is the same for any income >$30,000.