# understanding derivatives of a regression spline

I am trying to understand why regression splines are continuous at their knots

Suppose I am fitting a regression spline

$$E[Y|X] = \alpha + \beta_1 x + \beta_2 (x - t)^+$$

where $$(x - t)^+ = \max (x-t,0)$$.

Now, if I take the partial derivative of $$y$$ wrt $$x$$, I find

$$\frac{\partial y}{\partial x} = \begin{cases} \beta_1 \quad \text{if} \quad x \leq t,\\ \beta_1 + \beta_2 \quad \text{if} \quad x > t \end{cases}$$

For starters, is this correct?

If it is, I don't see how this shows the derivative at $$t$$ is continuous - it looks like there is a discrete change rather than smooth change