# How do I find one-sided limits of piecewise functions?

##### 1 Answer
May 16, 2016

If x = c is the junction for the two pieces $y = f \left(x\right) , x \in \left(a , c\right)$ and $y = g \left(x\right) , x \in \left(c , b\right)$, the one-sided limits at the junction x = c are $h \to 0$ of $f \left(c - h\right) \mathmr{and} g \left(c + h\right)$.

#### Explanation:

The piecewise functions are jointly continuous at x = c, if the one-sided limits are the same.

Further if the one-sided limits for f' ad g' are the same, they are differentiable at x = c.

For piece-wise cubic-spline fitting, this matching is extended up to f'' and g''.