Continuous-time signals are signals that are defined for every instant of time. They are often represented graphically as a continuous curve, where the horizontal axis represents time and the vertical axis represents the amplitude of the signal. Unlike discrete-time signals, which are defined only at specific intervals, continuous-time signals have values at every point in time.

Characteristics and Properties:

1. Amplitude: The value or strength of the signal at any given point in time.
2. Frequency: The rate at which the signal repeats its values.
3. Phase: The offset or delay of the signal relative to a reference point.

Mathematical Representation:
Continuous-time signals are typically represented by a function ( x(t) ), where ( t ) is a continuous variable denoting time.

Examples:

1. Sinusoidal Signal: ( x(t) = A \sin(2\pi f t + \phi) ), where ( A ) is the amplitude, ( f ) is the frequency, and ( \phi ) is the phase.
2. Exponential Signal: ( x(t) = Ae^{at} ), where ( a ) is a constant.
3. Unit Step Signal: ( x(t) = \begin{cases} 0, & \text{for } t < 0 \ 1, & \text{for } t \geq 0 \end{cases} )

Transformations:
Several mathematical transformations can be applied to continuous-time signals to analyze them in different domains:

1. Fourier Transform: Converts a signal from the time domain to the frequency domain.
2. Laplace Transform: Provides a complex frequency domain representation, which is particularly useful for solving differential equations.
3. Time Reversal: Reflects the signal about the vertical axis.
4. Time Scaling: Expands or compresses the signal in time.
5. Time Shifting: Delays or advances the signal in time.

Applications:
Continuous-time signals have various applications:

1. In analog electronics, like audio amplifiers and radio receivers.
2. In control systems, to understand the behavior of systems over time.
3. In communications, for modulating and demodulating analog signals.
4. In biomedical engineering, for signals like ECG and EEG.

Conversion to Discrete-Time Signals:
Continuous-time signals can be converted to discrete-time signals using a process called sampling. During sampling, the continuous signal is measured at regular intervals. The Nyquist-Shannon sampling theorem defines the conditions under which a continuous-time signal can be fully reconstructed from its samples.

In practice, while many real-world signals are continuous, they often get converted to discrete-time signals for processing in digital systems. However, the concepts and tools developed for continuous-time signals play a foundational role in understanding and analyzing the behavior of signals and systems.