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.