Continuity of piecewise functions Need help with this calculus worksheet. not exactly sure of what to do. I have an idea of how to do it but I don't know if it is right.
Suppose you have a definition of a piecewise function in the form
$$f(x)=\begin{cases}a(x) &amp;&amp; \text{if }x\geq 0\\b(x) &amp;&amp; \text{otherwise}\end{cases}$$
or something analogous, for continuous functions $a$ and $b$. If $f$ is continuous, then the limits $\lim_{x\rightarrow 0^+}f(x)$ and $\lim_{x\rightarrow 0^-}f(x)$ must agree. In particular, however,
$$\lim_{x\rightarrow 0^+}f(x)=\lim_{x\rightarrow 0^+}a(x)$$
since, as we approach $0$ from above, we will always be in the domain in which $f(x)=a(x)$. Similarly,
$$\lim_{x\rightarrow 0^-}f(x)=\lim_{x\rightarrow 0^-}b(x)$$
since, for $x$ less than $0$, it holds that $f(x)=b(x)$. Since $a$ and $b$ were assumed to be continuous, this is equivalent to saying that $a(0)=b(0)$.

This can be applied here, by considering, at each "transition" between one piece of the function to the next, whether the functions composing the part to the right and left of the boundary agree at the boundary.
![enter image description here](https://i.stack.imgur.com/bd63d.jpg.           X^3 if x &lt;0.       E^x if x>0

$$
  g(x) = 
\begin{cases}
  x^3 &amp; x \le 0 \\
  e^{x} x &gt; 0
\end{cases}
$$