A function q is defined for all real numbers by
q(x)=⎩⎨⎧cosx,mx+b,∣x−2∣+c,x<0,0≤x<2,x≥2,
where m, b, and c are real constants.
a. Part 1: Determine the conditions that m, b, and c must satisfy so that q is continuous at x=0 and at x=2. Express your answer as a system of equations and derive each equation using appropriate limit statements.
b. Part 2: Solve the system obtained in part (a) to find all ordered triples (m,b,c) that make q continuous on the entire real line.
c. Part 3: For one set of constants that you found in part (b), provide a complete, sentence-level proof that the corresponding q is continuous for all real x. Your proof must address each sub-interval and the junction points separately.