Say I have a piecewise-defined function
$$f(x) = \begin{cases} 2x+3 & \text{if}\; x \geq 1 \\ 2+3x & \text{if}\; x < 1 \end{cases}$$
What's the largest interval that this function can take to remain continuous? I am conflicted as to if its $(-\infty, 1)\cup[1,+\infty)$ or $(-\infty,+\infty)$. And is the function continuous with no discontinuities? Because if it is, then the largest interval is $(-\infty,+\infty)$.
Best Answer
Just check the continuity at $x=1$
So, $$f(1^-)=2+3*1=5$$ $$f(1)=2*1+3=5$$
So, $f$ is continuous everywhere i.e., $$x \in (-\infty,\infty)$$