Frequency spectrum of an amplitude modulated signal I am talking about a DSB- Suppressed Carrier amplitude modulation.
Let message signal be $m(t)=A_m \cos{\omega_c t}$ and carrier signal be $c(t)=A_c \cos{\omega_m t}$
Then the amplitude modulated signal is 
$$s(t)=m(t)c(t)$$
$$s(t)=A_c A_m \cos{\omega_c t} \cos{\omega_m t}$$
$$s(t)=\frac{A_c A_m}{2} \{\cos{(\omega_c+\omega_m) t} + \cos{(\omega_c-\omega_m) t}\} \tag3$$
In frequency domain $s(t)$ is

From eq 3 it is visible that two frequency components $\omega_c+\omega_m$ , $\omega_c-\omega_m$ are present.
Then in frequency domain $S(\omega)$ why isn't there an impulse at these locations ? (and at their negative frequencies, of course)
With $s$, $m$, and $c$ as you display, then the diagram of $S(\omega)$ is incorrect.  If it were, the spectrum of $m$ would be a triangle centered at zero frequency, which it is not.

$$
S(\omega) = \frac{A_cA_m}{4} \left[\\
\delta\left(\omega - (\omega_c + \omega_m)\right)\\
+ \delta\left(\omega - (\omega_c - \omega_m)\right)\\
+ \delta\left(\omega - (-\omega_c - \omega_m)\right)\\
+ \delta\left(\omega - (-\omega_c + \omega_m)\right)\\
\right]$$