Proving the continuity of a function using the definition

Question

Proving the continuity of a function using the definition. The function is: $f(x,y) = e^x + sen(y)$

I don’t know where to start. usually I have a point in where I have to find if the function is continuous, moreover, in this case I could just say that the function is continuous because is the composition of continuous functions, right?

Answer 1

$|f(x,y)-f(a,b)|=|e^x-e^a+\sin y-\sin b|\le e^a|e^{x-a}-1|+2|\cos\frac{y+b}2||\sin\frac{y-b}2|$ $\le e^a|e^{x-a}-1|+2|\sin\frac{y-b}2|$.

Given $\sqrt{(x-a)^2+(y-b)^2}\lt \delta$, we also have $|x-a|\lt \delta , |y-b|\lt \delta $. Hence in this case,

$|f(x,y)-f(a,b)|\lt e^a|e^{\delta}-1|+2|\sin\frac{\delta}2| \lt e^{a+\delta}+2|\sin\frac{\delta}2|\le 2\max\{e^{a+\delta},2\}$

i.e. for any $\varepsilon \gt 0$, one picks $\delta \gt 0$ such that $\varepsilon=2\max\{e^{a+\delta},2\}$

Thus the continuity of the function at the point $(a,b)$ and therefore everywhere in the domain of the function is proven.

(Note that if one uses the fact that $|\sin\frac{\delta}2|\le 1$, the modulus of continuity $\delta=\delta(\varepsilon)$ is obtained in a simpler way)

Answer 2

Let $a,b$ be any real numbers. Then $\lim_{x\rightarrow a, y\rightarrow b}e^x=e^a$ and $\lim_{x\rightarrow a, y\rightarrow b}\sin y=\sin b.$ Therefore $e^x$ and and $\sin y$ are continuous everywhere in the plane. Being a sum of two continuous functions, $f(x,y)$ is continuous everywhere. Similarly we can prove the above two functions continuous everywhere using $\epsilon-\delta $ definition as well. Can you try?