(a) Can $\sin(x+y)/(x+y)$ be made continuous by suitably defining it at (0,0)? (a) Can $\sin(x+y)/(x+y)$ be made continuous by suitably defining it at (0,0)? (b) Can $xy/(x^2 + y^2)$ be made continuous by suitably defining it at (0,0)? (c) Prove that $f: \mathbb R^2 \to \mathbb R$, $(x, y) \to ye^x + \sin(x) + (xy)^4$ is continuous. * * * Attempt (a) let $t = x + y$ $\lim_{(x, y) \to (0,0)} \frac{sin(x+y)}{x+y} = \lim_{t \to 0} \frac{sin(t)}{t} = 1$ is continuous at $0,0$ (b) Using $y = mx$ $\frac{x \cdot mx}{x^2 + (mx)^2} = \frac{m}{1+m^2}$ Not continuous since it's dependent on value of m (c) let $y = x, x = 0$ $\lim_{(x, y) \to (0,0)} ye^x + sin(x) + xy^4 = 0$ therefore continuous Am I right?

(a) You forgot to notice that the function is not defined on the line $x=-y$. But your reasoning outside of this subdomain is correct.

(b) Is correct

(c) You were asked to prove that the function is continuous **everywhere** , not that it it's limit is defined at $(0,0)$