Piecewise Functions: Graphs, Limits, Continuity and Derivatives

A piecewise function is a function defined by different formulas on different parts of its domain. The rule tells you which formula to use based on where x is. Piecewise functions are the standard test cases for limits, continuity and differentiability, because the interesting behavior happens where the pieces meet.

How to read a piecewise function

To evaluate, first find which condition x satisfies, then use that piece only. Here f(0) = 0^2 = 0, f(1) = 3 - 1 = 2 (because x \ge 1 includes 1), and f(2) = 3 - 2 = 1.

Graph of a piecewise function

Each piece is drawn only on its own interval. An open dot marks an endpoint that is not included; a closed dot marks one that is.

Limits of piecewise functions

At a break point a, compute the two one-sided limits separately, each with its own piece. For the function above at a = 1:

The one-sided limits differ, so \lim_{x\to 1}f(x) does not exist, and the graph has a jump. Away from the break point, a limit is just direct substitution into the active piece.

Continuity at a break point

f is continuous at a when all three of these agree:

1. the left-hand limit \lim_{x\to a^-}f(x),
2. the right-hand limit \lim_{x\to a^+}f(x),
3. the value f(a).

Example: choosing a constant. For which k is

continuous? The left limit at 2 is 2k + 1 and the right limit and value are 2^2 = 4. Setting 2k + 1 = 4 gives k = \frac{3}{2}.

Derivatives of piecewise functions

Inside each open interval, differentiate the active piece as usual. At a break point you need two checks:

• f must be continuous there (a jump means no derivative).
• The left and right derivatives must match: \lim_{h\to 0^-}\frac{f(a+h) - f(a)}{h} = \lim_{h\to 0^+}\frac{f(a+h) - f(a)}{h}.

When both pieces are differentiable polynomials and f is continuous at a, the second check reduces to comparing f_1'(a) with f_2'(a).

Example: a corner. The absolute value function |x| is -x for x < 0 and x for x \ge 0. It is continuous at 0, but the left slope is -1 and the right slope is 1, so |x| is not differentiable at 0.

Example: continuous but not smooth. For g above with k = \frac{3}{2}, the left slope at 2 is \frac{3}{2} and the right slope is 2x = 4. They differ, so g is continuous but not differentiable at x = 2.

Integrating a piecewise function

Split the integral at every break point inside the interval. For the first function on [0, 3]:

The second integral is \left[3x - \frac{x^2}{2}\right]_1^3 = \frac{9}{2} - \frac{5}{2} = 2. A single jump does not change the value of the integral, so it does not matter which piece owns the point x = 1.

Common mistakes

1. Using the wrong piece at the boundary. Check whether the condition says < or \le before evaluating f(a).
2. Concluding continuity from one side. Both one-sided limits and the value must agree.
3. Assuming continuous means differentiable. Corners like |x| at 0 are continuous but have no derivative.
4. Differentiating the pieces and comparing them at a jump. If f is not continuous at a, it is not differentiable there, even if the piece slopes happen to match.
5. Integrating one formula across the whole interval. Always split at the break points.

Related functions

The floor and ceiling functions are piecewise constant step functions with a jump at every integer, and absolute value is the simplest piecewise function with a corner. For the full treatment of limits and continuity, see the guide to continuity.