Question

For what values of a and b is continuous at every x?

f(x)=-2, x is less than or equal to -1
f(x)=ax-b, x is greater than -1 and less than 1
f(x)=3, x is greater than or equal to 1

Answer 1

`f(x)= \begin{cases} -2&x\le -1\\ ax-b&-1\ \textless \ x\ \textless \ 1\\ 3&x\ge 1 \end{cases}`

notice, is a piecewise
continuity for a piecewise means
one subfunction has to pick up where the previous one left off

so...if we look at the first subfunction
f(x) = -2 <-- that's a horizontal line up to the point of x = -1

so... whatever ax+b is, must be -2 also when x = -1

the subfunction following ax+b is
f(x) = 3 <--- another horizontal line, from the point x = 1 onwards

so... whatever ax+b is, must be 3 when x = 1
so f(x) = 3 picks up at that point

is that making any sense?