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Piecewise Continuity Checker

Check continuity of piecewise functions at boundary points

Define f(x) with two pieces: { a1*x+b1 for x<c, a2*x+b2 for x>=c }

f(x)={x+, x<

f(x)={x+, x>=2

Piecewise Continuity Test

1. Compute f(c) = a2*c+b2 (second piece)

2. Left limit: a1*c+b1 (first piece)

3. Right limit: a2*c+b2 (second piece)

4. Equal? Continuous at boundary

Checking continuity of piecewise functions requires evaluating the function and both one-sided limits at each boundary point. The function is continuous at the boundary if the left limit, function value, and right limit are all equal.

⚠ The function value at the boundary comes from the piece that includes the equality (x>=c). Adjust constants to make boundary continuous.

What Is Piecewise Continuity?

A piecewise function is continuous if each piece is continuous on its interval AND the function matches at the boundaries. At each boundary, the left piece value, function value, and right piece value must all be equal. If not, there is a jump or removable discontinuity.

Continuity Conditions

lim(x->c-)=lim(x->c+)=f(c). All three must be equal. f(c) defined by the piece that includes c.

Jump Discontinuity

Left and right limits are finite but not equal. The function jumps from one value to another at the boundary.

Removable (Fixing)

If left=right but f(c) differs, the discontinuity can be removed by redefining f(c) to match the limit.

Adjusting Parameters

To make continuous: set a1*c+b1 = a2*c+b2. Solve for the unknown constant. Choose b2 to match at boundary.

Teaching Example: f(x)={2x, x<2; 3x-2, x>=2}. At x=2: left=2*2=4, right=3*2-2=4, f(2)=3*2-2=4. All equal -> CONTINUOUS! Check: 2*2=4 and 3*2-2=4. Both pieces meet at (2,4).

Applications

Calculus Piecewise Models Signal Processing Physics Engineering

Frequently Asked Questions

How to check piecewise continuity?▼

At each boundary: compare left limit (from lower piece), right limit (from upper), and f(c). All equal -> continuous.

Jump vs removable?▼

Jump: left!=right (different values). Removable: left=right but f(c) differs. Jump can not be fixed by redefining f(c).

Which piece defines f(c)?▼

The piece with >= or >. In {x<2, x>=2}, the second piece x>=2 defines f(2). Check the inequality carefully.

How to make it continuous?▼

Set both pieces equal at boundary: a1*c+b1 = a2*c+b2. Solve for the unknown parameter.

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