Let $$f(x) = [2x^2 + 1]$$ and $$g(x) = \begin{cases} 2x - 3, & x < 0 \\ 2x + 3, & x \geq 0 \end{cases}$$, where $$[t]$$ is the greatest integer $$\leq t$$. Then, in the open interval $$(-1, 1)$$, the number of points where $$f \circ g$$ is discontinuous is equal to ______.

We have $$f(x) = [2x^2 + 1]$$ (where $$[\cdot]$$ is the greatest integer function) and $$g(x) = \begin{cases} 2x - 3, & x < 0 \\ 2x + 3, & x \geq 0 \end{cases}$$.

We need to find the number of points where $$f \circ g$$ is discontinuous in $$(-1, 1)$$.

For $$x \in (-1, 0)$$: $$g(x) = 2x - 3$$, so $$g(x) \in (-5, -3)$$. Then $$f(g(x)) = [2(2x-3)^2 + 1] = [2(4x^2 - 12x + 9) + 1] = [8x^2 - 24x + 19]$$.

As $$x$$ ranges over $$(-1, 0)$$, the expression $$h_1(x) = 8x^2 - 24x + 19$$ is a continuous function. At $$x = -1$$: $$h_1(-1) = 8 + 24 + 19 = 51$$. At $$x = 0$$: $$h_1(0) = 19$$. The minimum of $$h_1$$ on $$(-1, 0)$$ is at $$x = 0$$ (since vertex is at $$x = 3/2$$ which is outside this interval), so $$h_1$$ decreases from $$51$$ to $$19$$ on $$(-1, 0)$$.

The greatest integer function $$[h_1(x)]$$ is discontinuous whenever $$h_1(x)$$ passes through an integer. Since $$h_1$$ decreases continuously from $$51$$ to $$19$$, it passes through the integers $$50, 49, 48, \ldots, 20$$ — that is $$31$$ integer values. Each gives a point of discontinuity.

For $$x \in [0, 1)$$: $$g(x) = 2x + 3$$, so $$g(x) \in [3, 5)$$. Then $$f(g(x)) = [2(2x+3)^2 + 1] = [8x^2 + 24x + 19]$$.

Let $$h_2(x) = 8x^2 + 24x + 19$$. At $$x = 0$$: $$h_2(0) = 19$$. At $$x = 1$$: $$h_2(1) = 8 + 24 + 19 = 51$$. The function $$h_2$$ increases from $$19$$ to $$51$$ on $$[0, 1)$$, passing through integers $$20, 21, \ldots, 50$$ — that is $$31$$ integer values, each giving a discontinuity.

At $$x = 0$$, we check continuity. From the left: $$\lim_{x \to 0^-} [8x^2 - 24x + 19] = [19] = 19$$. From the right: $$f(g(0)) = [8(0) + 24(0) + 19] = [19] = 19$$. Since both sides give $$19$$, $$f \circ g$$ is continuous at $$x = 0$$. So $$x = 0$$ does not add a discontinuity.

The total number of discontinuities is $$31 + 31 = 62$$.