Continuity at the junction point

Piecewise-defined functions change formula at certain points. Precisely at these junction points continuity is not automatic and must be verified with the one-sided limits.

Remark — How to verify continuity at the junction point

Given $f(x) = \begin{cases} g(x) & x < x_0 \\ h(x) & x \ge x_0 \end{cases}$ the function is continuous at $x_0$ if and only if: $\lim_{x\to x_0^-}g(x) = \lim_{x\to x_0^+}h(x) = h(x_0).$ That is: the limit from the left (with the left-hand formula) must coincide with the limit from the right (with the right-hand formula) and with the value at $x_0$.

In practice one computes three quantities — left limit, right limit and value of the function at the point — and imposes that they all be equal. If parameters appear, this condition provides an equation that allows them to be determined.

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Topics: Continuita
Concepts: Continuita · Funzione definita a tratti · Limite destro e sinistro
Skills: Ragionare per casi