Continuity of a piecewise constant function 
A)I can draw the graph and see that the function is continuous at x=0.3 as when you approach it from the left and right you get the same result 
B) not sure how to prove properly but it is not continuous as you get different results from limit from right and left.
I have also put the solutions the teacher gave but they make no sense to me and are really hard to read/see 

 A: (a) To show $g\left(x\right)$ is continuous at $x=0.3$, choose any $\epsilon > 0$. We set $\delta = 0.1$.
Having $\vert x-0.3\vert < \delta$, means $0.2 \lt x \lt 0.4$. Now, $g\left(x\right) = 0$ for any $x$ in this interval, so we have:
$$\vert g\left(x\right) - g\left(0.3\right) \vert = 0 \lt \epsilon$$
(b) To show $g\left(x\right)$ is not continuous at $x = 0.5$ we need to find one $\epsilon \gt 0$ such that for any $\delta \gt 0$, there is always some $x$ such that $\vert x - 0.5 \vert \lt \delta$ for which: 
$$\vert g\left(x\right) - g\left(0.5\right) \vert \geq \epsilon$$
In other words, no matter how small we make our interval around $0.5$ on the $x$-axis (by shrinking $\delta$), there will be some value of $g\left(x\right)$ that doesn't get within $\epsilon$ of $g\left(0.5\right)$ on the $y$-axis.
So, we'll try it with $\epsilon = 0.1$. Consider any $\delta \gt 0$.
Having $\vert x - 0.5 \vert \lt \delta$, means $0.5 - \delta \lt x \lt 0.5 + \delta$. The value of $g\left(x\right)$ in this interval is $0$ or $1$. If $x = 0.5 - \delta/2$, say, we have
$$\vert g\left(x\right) - g\left(0.5\right) \vert = \vert 0-1 \vert = 1 \geq \epsilon$$
Therefore $g\left(x\right)$ is not continuous at $x = 0.5$.
