Calculus I · Limits and Continuity · lesson

Two-Parameter Continuity Problems

Two conditions and two parameters

Worked example

Solve a two-parameter continuity system

Choose $a$ and $b$ so that

f(x)= \begin{cases} ax+b,&x<1,\\ x^2+1,&1\le x<3,\\ 2x+a,&x\ge3 \end{cases}

is continuous at both $1$ and $3$ .

Show worked solution

At $x=1$ , continuity requires

a+b=1^2+1=2.

At $x=3$ , continuity requires

3^2+1=2(3)+a.

Thus,

10=6+a,

a=4.

Use $a+b=2$ :

4+b=2,

b=-2.

Therefore,

\boxed{a=4,\quad b=-2}.

After the explanation

Use the section idea

Reading lens

Do the limit, the function value, and the surrounding domain fit together at the point or across the interval?

Mental model

Continuity is a three-part agreement: the value exists, the two-sided limit exists, and those two quantities are equal.

Decision

At a point, test the three conditions in order; on an interval, check the domain and endpoints before invoking any continuity theorem.

Common trap

A sign change supports the Intermediate Value Theorem only when continuity holds on the entire closed interval, and it does not prove uniqueness.

Check yourself

You are ready to continue when you can classify a break, decide whether one value can repair it, and state every IVT hypothesis aloud.

Source & rights

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