Hey this might be a dumb question so here it goes:
Is $e^{(x)}$ continuous for $x\in \mathbb{C}$?
Specifically this question arose while solving the differential equation in the form of
$\dfrac{dy}{dx}=\dfrac{y}{x}$ with the solution $y=cx$ and was curious if it was possible for $c=0$ in the context of complex numbers.
Thanks and be gentle, I'm an engineering student.
On
If you know that if $x,y$ are real and $e^{x+iy} = e^{\sqrt{x^2+y^2}}(\cos y + i\sin y)$ and that the functions involved in putting that last expression together, you can deduce that that function of $x+iy$ is continuous (provided you know several basic facts about continuity).
If you know that $\displaystyle e^z = \sum_{n=0}^\infty\frac{z^n}{n!}$, and that, as shown by a ratio test, that series converges, then you can deduce that $z\mapsto e^z$ is continuous if you know some basic theorems on convergent power series.
If you know that $z\mapsto e^z$ is differentiable and that differentiable functions are continuous, that also does it.
Yes it is continuous. It is well-known that $f(z)=e^z$ is even holomorphic, which means it's complex differentiable.