Can there be a bump function with bounds on some partial derivatives?

I am looking for a smooth function $$h:\mathbb{R}^2\to\mathbb{R}$$ such that,

1. $$Supp(h)\subset B(0,1)$$
2. $$|h(0)| \ge 1$$
3. $$|\partial_x h| < \epsilon$$ everywhere, where $$\epsilon > 0$$ is a given constant

I have seen this and this, where the answer is negative for bounded derivatives. But note that I don't require any control over the partial derivative $$\partial_y h$$.

So can such a function $$h$$ exist? Or is the situation still the same and we cannot expect any kind of derivative bound at all?

If it helps, geometrically I am trying to think of climbing a hill of height $$1$$ and base $$B(0,1)$$, by using a road with many hairpin bends : as I am traversing the road the inclination (i.e $$\partial_x h$$) doesn't grow too much; but as I am crossing a bend, the $$\partial_y h$$ value gets arbitrarily large. I apologize if this makes no sense at all!

Any help is appreciated. Cheers!

For any $${\bf a}, {\bf b} \in \mathbb R^2$$, and any polygonal path $$\Gamma$$ from $$\bf a$$ to $$\bf b$$, $$h({\bf b}) - h({\bf a}) = \int_\Gamma (\nabla h)({\bf r}) \cdot d{\bf r}$$
In particular, take $${\bf a} = (-1,0)$$ and $${\bf b} = (0,0)$$, and $$\Gamma$$ the straight line segment from $$\bf a$$ to $$\bf b$$. We get

$$1 \le h({\bf b}) - h({\bf a}) = \int_{-1}^0 \partial_x h(t,0)\; dt \le \max_{0 \le t \le 1} \partial_x h(t,0)$$

i.e. there must be some $$t \in [0,1]$$ with $$\partial_x h(t,0) \ge 1$$.

The point is that the first equation is true for every polygonal path, not just for some. You don't get to choose the road!