Discuss the continuity of the following functions:
a) $f(x) = \sin x + \cos x$
b) $f(x) = \sin x - \cos x$
c) $f(x) = \sin x \times \cos x$

(A) It is known that if $g$ and $h$ are two continuous functions,then $g+h$,$g-h$,and $g \cdot h$ are also continuous.
First,we prove that $g(x) = \sin x$ and $h(x) = \cos x$ are continuous functions.
For $g(x) = \sin x$:
$g(x)$ is defined for every real number. Let $c$ be a real number. Put $x = c + h$. As $x \to c$,$h \to 0$.
$g(c) = \sin c$.
$\lim_{x \to c} g(x) = \lim_{h \to 0} \sin(c + h) = \lim_{h \to 0} (\sin c \cos h + \cos c \sin h) = \sin c(1) + \cos c(0) = \sin c$.
Since $\lim_{x \to c} g(x) = g(c)$,$g(x)$ is continuous.
For $h(x) = \cos x$:
$h(x)$ is defined for every real number. Let $c$ be a real number. Put $x = c + h$. As $x \to c$,$h \to 0$.
$h(c) = \cos c$.
$\lim_{x \to c} h(x) = \lim_{h \to 0} \cos(c + h) = \lim_{h \to 0} (\cos c \cos h - \sin c \sin h) = \cos c(1) - \sin c(0) = \cos c$.
Since $\lim_{x \to c} h(x) = h(c)$,$h(x)$ is continuous.
Conclusion:
a) $f(x) = g(x) + h(x) = \sin x + \cos x$ is continuous.
b) $f(x) = g(x) - h(x) = \sin x - \cos x$ is continuous.
c) $f(x) = g(x) \cdot h(x) = \sin x \cos x$ is continuous.