Abed is studying data science. Abed tries to develop a machine model

learning to predict carbon dioxide emissions based on daily temperature. Abed uses the sigmoid function to normalize the temperature data, which is expressed as follows:

S$($x$)=1/1+$ e$^($x$)$

Apart from that, Abed also uses the hyperbolic tangent $($tanh$)$ activation function in neural

artificial network, which is stated as follows:

T$($x$)=($e$^$x $$ e$^(-$x$))/($e$^$x $+$ e$^(-$x$))$

During model development, Abed discovered that the model performance was very sensitive to extreme temperature values $$causing the sigmoid function values $$to approach $0$ or $1$ and the value of the tanh function approaches $1$ or $1.$ Abed wants to ensure that these functions are continuous. To that end, answer the following questions to help Abed in developing the model!

a$)$ Prove that the sigmoid function S$($x$)$ and the activation function T$($x$)$ are continuous for all x in R

b$)$ Abed considers using the ReLU $($Rectified Linear Unit$)$ function

which is expressed as R$($x$)=$ max$(0,$ x$).$ Is ReLU function continuous? Explain your answer!