We know by Leibniz's Rule (as stated, for instance, on p. 324 of Munkres'"Analysis on Manifolds", Westview Press 1991) that, given two compact intervals $I = [a,b], J=[c,d] \subseteq \mathbb{R}$ with non-empty interiors, and a function $f\in I\times J\mapsto \mathbb{R}$ that is continuous on $I\times J$, the function $g\in I \mapsto \mathbb{R}$ defined for every $x \in I$ thus: $$ g(x) := \int_c^d f(x,y)\ dy, $$ is continuous on $I$.
Are there any conditions on $f$, which are less restrictive than continuity everywhere in $I\times J$, that ensure continuity, or even almost-everywhere continuity, of $g$?
The Short Answer
If $f$ is Riemann-integrable on $I\times J$, and if, additionally, for every $x \in I$, the function $y\in J \rightarrow f(x,y)$ is Riemann-integrable on $J$, then $g$ is Riemann-integrable on $I$. Also recall that a real-valued function $\varphi$ defined on a compact interval $K\subseteq\mathbb{R}$ with non-empty interior is Riemann-integrable iff it is bounded uniformly on $K$ and continuous Lebesgue-almost-everywhere on $K$, and note that this characterization can be extended to $\mathbb{R}^2$.
In what follows I shall state and prove these results with greater precision and generality (see Conclusion 15 below).
The Long Answer
Definition 1 $\mathcal{I} := \{[a,b]\ :\!|\ a,b\in\mathbb{R},\ a<b\}$.
Definition 2 For every $n \in \{1, 2, \dots\}$ we define $$ \mathcal{I}_n := \left\{I_1\times\cdots\times I_n\ :\!|\ I_1, \dots, I_n \in \mathcal{I}\right\}. $$
Definition 3 We define a function $\ell:\mathcal{I}\rightarrow\mathbb{R}$ as follows. For every $[a,b] \in \mathcal{I}$, $\ell\left([a,b]\right) := b-a$.
Definition 4 For every $n \in \{1, 2, \dots\}$, we define the function $v_n:\mathcal{I}_n\rightarrow\mathbb{R}$ as follows. For every $I_1, I_2, \dots, I_n \in \mathcal{I}$, $$ v_n(I_1\times I_2 \times \cdots \times I_n) := \prod_{k = 1}^n\ell(I_k). $$
Definition 5 For every $n \in \{1, 2, \dots\}$, denote by $\mathcal{E}_n$ the Euclidean topology on $\mathbb{R}^n$, i.e. the topology induced by the Euclidean norm on $\mathbb{R}^n$.
Definition 6 For every $n \in \{1,2,\dots\}$, denote by $\mathcal{M}_n$ the $\sigma$-algebra of Lebesgue-measurable subsets of $\mathbb{R}^n$.
Definition 7 For every $n \in \{1, 2, \dots\}$, denote by $\lambda_n$ the Lebesgue-measure on $\mathcal{M}_n$.
Definition 8 For every $n \in \{1, 2, \dots\}$, for every non-empty $D \subseteq \mathbb{R}^n$, and for every $f\in D\rightarrow \mathbb{R}$, denote by $\mathbf{d}_n(f, D)$ the set of points $x \in D$, where $f$ is not continuous w.r.t. the following topologies: on the domain - the topology induced on $D$ by $\mathcal{E}_n$, on the range - $\mathcal{E}_1$.
Definition 9 Let $n \in \{2, 3, \dots\}$. For every $I_1, I_2, \dots, I_n \in \mathcal{I}$, for every $f\in I_1\times I_2\times\cdots\times I_n \rightarrow \mathbb{R}$, and for every $m \in \{1, \dots, n-1\}$, denote by $\mathfrak{c}_n(f,m)$ the result of Currying $f$ in the following fashion. Notate $J:=I_1\times\cdots\times I_m$, $K:=I_{m+1}\times\cdots\times I_n$. Then $\mathfrak{c}_n(f,m)$ is defined to be the function $\mathfrak{c}_n(f,m) \in J \rightarrow (K\rightarrow \mathbb{R})$, which, to every $(x_1, \dots, x_m) \in J$ assigns the function $\mathfrak{c}_n(f,m)(x_1, \dots, x_m) \in K\rightarrow\mathbb{R}$, which to every $(x_{m+1}, \dots, x_n) \in K$ assigns $$ \mathfrak{c}_n(f,m)(x_1, \dots, x_m)(x_{m+1},\dots, x_n) := f(x_1, \dots, x_n). $$
Definition 10 For every $n \in \{1, 2, \dots\}$ denote by $\mathcal{R}_n$ the operator, which, to every $I \in \mathcal{I}_n$ assigns the set of functions $f\in I\rightarrow\mathbb{R}$ that are Riemann-integrable on $I$.
Definition 11 For every $n \in \{1, 2, \dots\}$ we denote by $\rho_n$ the operator of Riemann-integration, namely given an interval $I \in \mathcal{I}_n$, $\rho_n(I)$ is a function $\rho_n(I) \in \mathcal{R}_n(I)\rightarrow\mathbb{R}$ that assigns to every $f\in\mathcal{R}_n(I)$ the Riemann integral $$ \rho_n(I)(f) := \mathrm{(R)}\hspace{-.25cm}\int_I\ f. $$
Theorem 12 Let $n \in \{1, 2, \dots\}$, let $I \in \mathcal{I}_n$, and let $f\in I\rightarrow \mathbb{R}$. Then $f \in \mathcal{R}_n(I)$ iff the following three conditions hold:
a. $f$ is bounded on $I$,
b. $\mathbf{d}_n(f,I) \in \mathcal{M}_n$,
c. $\lambda_n\left(\mathbf{d}_n(f,I)\right) = 0$.
Proof This is the content of Theorem 1, "Lebesgue's Criterion", on p. 111 of Zorich's "Mathematical Analysis II" (Springer 2000). Q.E.D.
Lemma 13 Let $n \in \{2, 3, \dots\}$, let $m \in \{1, \dots, n-1\}$, let $I_1, I_2, \dots, I_n \in \mathcal{I}$, and let $f\in I_1\times I_2\times\cdots\times I_n\rightarrow\mathbb{R}$. Notate $$ \begin{align} I &:= I_1\times\cdots\times I_n,\\ J &:= I_1\times\cdots I_m,\\ K &:=I_{m+1}\times\cdots\times I_n. \end{align} $$
Let $M \in [0,\infty)$ be such that, for every $z \in I$, $|f(z)|\leq M$.
Denote by $M_K$ the constant function $M_K \in K\rightarrow\mathbb{R}$, which to every $y \in K$ assigns $M_K(y) := M$. Denote by $\mathrm{abs}$ the function $\mathrm{abs}:\mathbb{R}\rightarrow\mathbb{R}$, which to every $r \in \mathbb{R}$ assigns the absolute value $\mathrm{abs}(r) := |r|$.
Then
a. For every $x \in J$, $\mathrm{abs}\circ\mathfrak{c}_n(f,m)(x)\leq M_K$.
b. Suppose, additionally, that, for every $x \in J$, $\mathfrak{c}_n(f,m)(x) \in \mathcal{R}_{n-m}(K)$. Then, for every $x \in J$, $$ \left|\mathrm{(R)}\hspace{-.25cm}\int_K \mathfrak{c}_n(f,m)(x)\right| \leq M v_n(K). $$
Proof
a. Let $x=(x_1, \dots, x_m) \in J$. Then, for every $y = (x_{m+1}, \dots, x_n)\in K$, $$ \left(\mathrm{abs}\circ\mathfrak{c}_n(f,m)(x)\right)(y) = \left|\mathfrak{c}_n(f,m)(x)(y)\right| = |f(x_1, \dots, x_n)| \leq M = M_K(y). $$
b. This is the content of Theorem 10.4 on p. 87 of Munkres' "Analysis on Manifolds" (Westview Press 1991).
Q.E.D.
Theorem 14 Let $n \in \{2, 3, \dots\}$, let $m \in \{1, \dots, n-1\}$, let $I_1, I_2, \dots, I_n \in \mathcal{I}$, and let $f\in I_1\times I_2\times\cdots\times I_n\rightarrow\mathbb{R}$. Notate $$ \begin{align} I &:= I_1\times\cdots\times I_n,\\ J &:= I_1\times\cdots I_m,\\ K &:=I_{m+1}\times\cdots\times I_n. \end{align} $$ Suppose that the following two conditions hold:
$f \in \mathcal{R}_n(I)$,
For every $x \in J$, we have $\mathfrak{c}_n(f,m)(x) \in \mathcal{R}_{n-m}(K)$.
Then $\rho_{n-m}(K)\circ\mathfrak{c}_n(f,m) \in \mathcal{R}_m(J)$.
Proof This is a part of Theorem 200.1, "Satz von Fubini", on p. 450 of Heuser's "Lehrbuch der Analysis, Teil 2", 13th edition (Teubner 2004). Q.E.D.
Conclusion 15 Let $n \in \{2, 3, \dots\}$, let $m \in \{1, \dots, n-1\}$, let $I_1, I_2, \dots, I_n \in \mathcal{I}$, and let $f \in I_1\times I_2\times\cdots\times I_n \rightarrow \mathbb{R}$. Notate $$ \begin{align} I &:= I_1\times\cdots\times I_n,\\ J &:= I_1\times\cdots I_m,\\ K &:= I_{m+1}\times\cdots\times I_n. \end{align} $$
Suppose that the following two conditions hold:
$f$ is bounded on $I$, $\mathbf{d}_n(f, I) \in \mathcal{M}_n$ and $\lambda_n\left(\mathbf{d}_n(f, I)\right) = 0$,
For every $x \in J$, $\mathbf{d}_{n-m}\left(\mathfrak{c}_n(f,m)(x), K\right) \in \mathcal{M}_{n-m}$ and $\lambda_{n-m}\left(\mathbf{d}_{n-m}\left(\mathfrak{c}_n(f,m)(x), K\right)\right) = 0$.
Then
a. $\rho_{n-m}(K)\circ\mathfrak{c}_n(f,m)$ is bounded on $J$.
b. $\mathbf{d}_m\left(\rho_{n-m}(K)\circ\mathfrak{c}_n(f,m),J\right) \in \mathcal{M}_m$.
c. $\lambda_m\left(\mathbf{d}_m\left(\rho_{n-m}(K)\circ\mathfrak{c}_n(f,m),J\right)\right) = 0$.
Proof
In light of condition (1), Theorem 12 implies Theorem 14(1).
Since, by condition (1), $f$ is bounded on $I$, Lemma 13(a) implies that, for every $x \in J$, $\mathfrak{c}_n(f,m)$ is bounded on $K$. In light of this fact and of condition (2), Theorem 12 implies Theorem 14(2).
Since both conditions of Theorem 14 hold, its conclusion holds too.
Conclusions (a), (b) and (c) now follow from conclusions (a), (b) and (c), respectively, of Theorem 12.
Q.E.D.