Are piecewise functions continuous?

Question

Determine which of three $2\pi$-periodic functions are continuous:

$f_1(x)= \begin{cases} 0 & \text{for $x\in[0,\pi[$} \\ 4\pi-x & \text{for $x\in[3\pi,4\pi[$} \\ \end{cases}$

$f_2(x)= \begin{cases} \cos(\frac{x}{2}) & \text{for $x\in[0,\pi[$} \\ \frac{1}{\pi^2}(x-3\pi)^2 & \text{for $x\in[3\pi,4\pi[$} \\ \end{cases}$

$f_3(x)= \begin{cases} \sin(\frac{x}{2}) & \text{for $x\in[0,\pi[$} \\ -\cos(x) & \text{for $x\in[3\pi,4\pi[$} \\ \end{cases}$

Attempt

By definition, a function is continuous if and only if $\lim_{x\rightarrow x_0}f(x)=f(x_0) $ for all points in the functions range.

I tried plotting the three piecewise function to see for which the above definition holds, and it seems like it's only $f_2(x)$ that follows the definition. However, the function still has "gaps" and I'm therefore in doubt if it actually is a continuous function. Can someone clarify this for me?

Here are the plots: -

Answer 1

You're failing to account for the fact that $f_1$, $f_2$, $f_3$ all have periods of $2\pi$. As such, each of the functions is defined on all of $\mathbb R$. Nevertheless, your question opens up a point I think is important to address.

By your definition of continuity, none of your plotted functions are continuous. This is because in order for a limit $\displaystyle{\lim_{x\to x_0}f(x)}$ to exist, the function must be defined in some open interval containing $x_0$. This won't happen in any of your functions at $x_0=\pi$.

However, there are other definitions of continuity, whereby a function is also continuous at $x_0$ if either one-sided limit exists, and the function is undefined on the other side. Under these definitions, all three of your plotted functions would be continuous.

So, the answer to "which of the functions corresponding to these three plots is continuous?" can be either "none" or "all" depending on the fine points of your definition of continuity. As to the original question, I'll leave that up to you.

Answer 2

$f(x)$ is continuos at $x=c$ given that: $$ \lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x) = \lim_{x\to c} f(x) = f(c) $$ Hence, $f(c)$ must be defined. $f_2(x)$ is not defined for every real number $x$ so it is not a continuous function.