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What are some functions that satisfy:

1) $x(t) \to y(t)$ and then $ax(t)\to ay(t)$

but not this property:

2) If $x_1(t) \to y_1(t)$ and $x_2(t) \to y_2(t)$ then $x_1(t) + x_2(t) \to y_1(t) + y_2(t)$

I cannot think of any. I considered $y = x^2$, but that doesn't satisfy the first property. Nothing else seems possible. Also, this question came from my professor, but Wikipedia says that homogeneity is defined to be: If $x(t) \to y(t)$ then $ax(t) \to a^ky(t)$ which would make more sense to me.

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    Your notation is not quite clear. Do you mean that for function $f$, 1) $f(a\cdot x)=a\cdot f(x)$ and 2) $f(x+y)=f(x)+f(y)$? And what is the domain for $a$: complex numbers, real numbers, rational numbers, integers, other? What is the domain of the function $f$? Those domains matter. – 2017-01-18
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    @RoryDaulton Yes that is what I mean. I believe the the domain is real numbers – 2017-01-18
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    Is the domain of function $f$ also the real numbers? Remember that other possible domains include $\mathbb R^n$ for any positive integer $n$. – 2017-01-18

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Here is an example function from $\mathbb R^2$ to $\mathbb R$ that satisfies homogeneity,

$$f(a\cdot x)=a\cdot f(x)$$

for all $a\in\mathbb R$, $x\in\mathbb R^2$, but does not satisfy additivity,

$$f(x+y) = f(x)+f(y)$$

for all $x,y\in\mathbb R^2$.

Define

$$f((r,s)) = \min(r, 2s)$$

Then it is easy to see that this function satisfies homogeneity, but letting $x=(1,0)$, $y=(1,1)$ we see that

$$f(x)=\min(1,2\cdot 0)=\min(1,0)=0$$ $$f(y)=\min(1,2\cdot 1)=\min(1,2)=1$$ but $$f(x+y)=f((2,1))=\min(2,2\cdot 1)=\min(2,2)=2$$

So here, $f(x+y)\ne f(x)+f(y)$.

Note that the basis for this example is in a section of the very Wikipedia article that you reference in your question.

If the $a$ range over the rationals, I can think of a similar example of a real-valued function of real numbers, but it is more complicated to explain.