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Continuity of a Piecewise Function

Continuity and Differentiability · Class XII

Piecewise: f(x) = x + a for x < 0 and bx + 1 for x ≥ 0. Tune a, b until the left and right limits match.

1
type a value
-33
1
type a value
-33
Live values
  • Left limit at 01
  • Right limit at 01
  • Jump0
xy
  • Piecewise f(x)

Formulas in this lab

  • Left limit at 0
    lim⁡x→0−f=a\lim_{x\to 0^-} f = a
  • Right limit at 0
    lim⁡x→0+f=1\lim_{x\to 0^+} f = 1
  • Jump
    1−a1 - a
Tip: Continuous at 0 ⇔ a = 1. Differentiable ⇔ a = 1 and b = 1.

Frequently asked questions

▶When is a piecewise function continuous at the join?

At the join $x = c$, the function is continuous iff $\lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x) = f(c)$. For $f(x) = x + a$ ($x < 0$) and $f(x) = bx + 1$ ($x \ge 0$), the join at $0$ requires $a = 1$.

▶How do I use the piecewise continuity lab?

Slide $a$ and $b$. The lab plots both branches and reports left limit ($a$), right limit ($1$) and the jump ($1 - a$). Set $a = 1$ to make the two pieces meet smoothly at $x = 0$.

▶Continuity versus differentiability: what's the extra requirement?

Continuity needs left and right limits to match the value. Differentiability also needs left and right slopes to match: $f'(c^-) = f'(c^+)$. Here, that means $a = 1$ (continuous) and $b = 1$ (slope $1$ on both sides). Continuous but not differentiable is a JEE favourite trap.

▶Where do piecewise functions appear in real life?

Income tax slabs: tax depends piecewise on income. Telecom tariffs (different rates per slab of minutes), step-function depreciation in accounting, and stress-strain curves with elastic-plastic transitions all use piecewise functions. Continuity matters wherever a real system shouldn't have sudden jumps.