\begin{align*} f: (0,2] &\to \mathbb{C},\\ x &\mapsto x^2 \sin(\pi/x) + ix^2, \quad x \in (0, 1]\\ x &\mapsto \alpha e^\frac{1}{1-x}+i\beta e^x. \quad x \in (1, 2] \end{align*}
I need to show that $f$ is continuous on $(0, 1)$ and $(1, 2]$ which I managed to do using sequences. Then I need to determine whether $\alpha$ and $\beta$ can be chosen to make $f$ continuous on $(0, 2]$. So I need to check if it's possible to pick $\alpha$ and $\beta$ in order to make $$\lim_{x\to 1^-} f(x)=f(x) = \lim_{x\to 1^+}f(x),$$ which means f is continuous at $1$.
The problem is: $f(x) = \alpha e^\frac{1}{1-x}+i\beta e^x$ would result in divison by zero when $\lim_{x\to 1} f(x)$.
I thought about picking $\alpha=\frac{-1}{e^\frac{1}{1-x}}+\frac{1}{e^\frac{1}{1-x}}$ and $\beta=\frac{1}{e}$, because then $$f(x) = \alpha e^\frac{1}{1-x}+i\beta e^x = \left(\frac{-1}{e^\frac{1}{1-x}}+\frac{1}{e^\frac{1}{1-x}}\right) e^\frac{1}{1-x}+\frac{i}{e} e^x = (-1+1)+\frac{i}{e}e^x$$ and therefore $f(1)= i$.
The last question is if $f$ can be made continuous on all of $\mathbb{R}$ and if yes, how?
I'm rather sure that my idea won't work, so I would appreciate your help!