Continuity of multivariable functions by looking at single variable components

I am aware that sums, products and quotients of continuous multivariable functions gives continuous multivariable functions. My query is whether knowing that some constituent single variable function means the overall multivariable function is continuous.

The example I am currently working on is proving that the function given by:

f(x,y)=(sin(x-y))/(x-y) where x=/=y and f(x,y) = 1 when x=y is continuous everywhere

I have expanded the sin(x-y) into a product of single variable trig functions.

Since f is then a combination of continuous single variable functions does that mean the multivariable function is continuous?

If no would anyone be able to give me a hint as to how to use either the epsilon-delta or sequential definition to show that f is continuous everywhere.

• How did you expand $\sin(x-y)$ into a product?
– zhw.
Oct 6, 2018 at 18:45
• Sin(x-y)=sin(X)cos(y)-sin(y)cos(X) Oct 7, 2018 at 8:26
• That's not a product, it's a difference of products.
– zhw.
Oct 7, 2018 at 15:41

The function $$g(u) = (\sin u)/u, u\ne 0,$$ $$g(0)=1,$$ is continuous on $$\mathbb R.$$ Your function $$f(x,y)$$ equals $$g(x-y)$$ for all $$(x,y).$$ Thus $$f$$ is the composition of continuous functions, hence is continuous.
• $g$ is continuous everywhere except possibly at $0.$ But at $0$ we know $g(u)\to 1 = g(0).$ I'm not sure what you are doing with those derivatives.