I have a number of functions: $[x], [x]+[-x],[x]+\{x\}^2, [x]+\{x\}^{1+|x|}, \frac{[x]}{1+\{x\}},[x]\sin(\pi x)$. I need to give the continuous functions for all $x \in \mathbb{R}$. And apparently my answers are wrong. $[x]$ is the whole part of the number. For example, $[0.1] = 0$,$[1.1] = 1$,$[-0.1] = -1$, $[-1.1]=2$. And $\{x\} = x - [x]$ is the fractional part of the number.
My thinking in this problem was that I simply look at the discontinuous points for these functions and see if they are in fact continuous. That sounds a bit weird. What I mean is that we know that there are "jumps" in $[x]$ at $x$ which are natural numbers, for example. This thinking led me to the following conclusions for the above set of functions:
$$\text{(i) } [x] + [-x] \text { is continuous} \forall x \\ \lim_{x \to 1}f(x) = f(1) = 0 \\ \text{ in fact it is } 0 \ \forall \ x$$
$$\text{(ii) } [x] + \{x\}^2 \text{ is continuous } \forall x \\ \text{ consider } \lim_{x \to 0} f(x) = f(0) = 0$$
$$\text{(iii) } [x] + \{x\}^{1+|x|} \text{ is continuous } \forall x \\ \text{ consider } \lim_{x \to 0} f(x) = f(0) = 0$$
The other three are discontinuous. But apparently I am making a mistake somewhere.
Discontinuous at some $x$:
$$\text{(iv) } \frac{[x]}{1+\{x\}} \text{ is not continuous } \forall x \\ \text{ consider } \lim_{x \to 0} f(x) = -\frac{1}{2} \neq f(0) = 0$$
$$\text{(v )} [x] \text{ trivial }$$
$$\text{(vi )} [x]\sin(\pi x) \text{ is not continuous because } [x] \\ \text{is not}$$