Prove that the function \(f(x)=5x-3\) is continuous at \(x=0\), at \(x=-3\) and at \(x=5\).
Examine the continuity of the function \(f(x)=2x^2-1\) at \(x=3\).
\(f\) is continuous at \(x=3\).
Examine the following functions for continuity:
(a) \(f(x)=x-5\)
(b) \(f(x)=\frac{1}{x-5},\; x\ne 5\)
(c) \(f(x)=\frac{x^2-25}{x+5},\; x\ne -5\)
(d) \(f(x)=|x-5|\)
(a), (b), (c) and (d) are all continuous functions.
Prove that the function \(f(x)=x^n\) is continuous at \(x=n\), where \(n\) is a positive integer.
Is the function \(f\) defined by
\[ f(x)=\begin{cases} x, & x\le 1 \\ 5, & x>1 \end{cases} \]
continuous at \(x=0\)? At \(x=1\)? At \(x=2\)?
\(f\) is continuous at \(x=0\) and \(x=2\); not continuous at \(x=1\).
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} 2x+3, & x\le 2 \\ 2x-3, & x>2 \end{cases} \]
Discontinuous at \(x=2\).
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} |x|+3, & x\le -3 \\ -2x, & -3
Discontinuous at \(x=3\).
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} \frac{|x|}{x}, & x\ne 0 \\ 0, & x=0 \end{cases} \]
Discontinuous at \(x=0\).
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} \frac{x}{|x|}, & x<0 \\ -1, & x\ge 0 \end{cases} \]
No point of discontinuity.
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} x+1, & x\ge 1 \\ x^2+1, & x<1 \end{cases} \]
No point of discontinuity.
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} x^3-3, & x\le 2 \\ x^2+1, & x>2 \end{cases} \]
No point of discontinuity.
Find all points of discontinuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} x^{10}-1, & x\le 1 \\ x^2, & x>1 \end{cases} \]
\(f\) is discontinuous at \(x=1\).
Is the function defined by
\[ f(x)=\begin{cases} x+5, & x\le 1 \\ x-5, & x>1 \end{cases} \]
a continuous function?
\(f\) is not continuous at \(x=1\).
Discuss the continuity of the function \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} 3, & 0\le x\le 1 \\ 4, & 1
\(f\) is not continuous at \(x=1\) and \(x=3\).
Discuss the continuity of the function \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} 2x, & x<0 \\ 0, & 0\le x\le 1 \\ 4x, & x>1 \end{cases} \]
\(x=1\) is the only point of discontinuity.
Discuss the continuity of the function \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} -2, & x\le -1 \\ 2x, & -1
Continuous
Find the relationship between \(a\) and \(b\) so that the function \(f\) defined by
\[ f(x)=\begin{cases} ax+1, & x\le 3 \\ bx+3, & x>3 \end{cases} \]
is continuous at \(x=3\).
\(a=b+\frac{2}{3}\)
For what value of \(\lambda\) is the function defined by
\[ f(x)=\begin{cases} \lambda(x^2-2x), & x\le 0 \\ 4x+1, & x>0 \end{cases} \]
continuous at \(x=0\)? What about continuity at \(x=1\)?
For no value of \(\lambda\), \(f\) is continuous at \(x=0\) but \(f\) is continuous at \(x=1\) for any value of \(\lambda\).
Show that the function defined by \(g(x)=x-[x]\) is discontinuous at all integral points. Here \([x]\) denotes the greatest integer less than or equal to \(x\).
Is the function defined by \(f(x)=x^2-\sin x+5\) continuous at \(x=\pi\)?
\(f\) is continuous at \(x=\pi\).
Discuss the continuity of the following functions:
(a) \(f(x)=\sin x+\cos x\)
(b) \(f(x)=\sin x-\cos x\)
(c) \(f(x)=\sin x\cdot\cos x\)
(a), (b) and (c) are all continuous
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Cosine function is continuous for all \(x\in\mathbb{R}\); cosecant is continuous except for \(x=n\pi,\; n\in\mathbb{Z}\); secant is continuous except for \(x=\frac{(2n+1)\pi}{2},\; n\in\mathbb{Z}\) and cotangent function is continuous except for \(x=n\pi,\; n\in\mathbb{Z}\).
Find all points of discontinuity of \(f\), where
\[ f(x)=\begin{cases} \frac{\sin x}{x}, & x<0 \\ x+1, & x\ge 0 \end{cases} \]
There is no point of discontinuity.
Determine if \(f\) defined by
\[ f(x)=\begin{cases} x^2\sin\frac{1}{x}, & x\ne 0 \\ 0, & x=0 \end{cases} \]
is a continuous function.
Examine the continuity of \(f\), where \(f\) is defined by
\[ f(x)=\begin{cases} \sin x-\cos x, & x\ne 0 \\ -1, & x=0 \end{cases} \]
\(f\) is continuous for all \(x\in\mathbb{R}\).
Find the value of \(k\) so that the function \(f\) is continuous at \(x=\frac{\pi}{2}\), where
\[ f(x)=\begin{cases} \dfrac{k\cos x}{\pi-2x}, & x\ne \dfrac{\pi}{2} \\ 3, & x=\dfrac{\pi}{2} \end{cases} \]
\(k=6\)
Find the value of \(k\) so that the function \(f\) is continuous at \(x=2\), where
\[ f(x)=\begin{cases} kx^2, & x\le 2 \\ 3, & x>2 \end{cases} \]
\(k=\frac{3}{4}\)
Find the value of \(k\) so that the function \(f\) is continuous at \(x=\pi\), where
\[ f(x)=\begin{cases} kx+1, & x\le \pi \\ \cos x, & x>\pi \end{cases} \]
\(k=-\frac{2}{\pi}\)
Find the value of \(k\) so that the function \(f\) is continuous at \(x=5\), where
\[ f(x)=\begin{cases} kx+1, & x\le 5 \\ 3x-5, & x>5 \end{cases} \]
\(k=\frac{9}{5}\)
Find the values of \(a\) and \(b\) such that the function defined by
\[ f(x)=\begin{cases} 5, & x\le 2 \\ ax+b, & 2 is a continuous function.
\(a=2,\; b=1\)
Show that the function defined by \(f(x)=\cos(x^2)\) is a continuous function.
Show that the function defined by \(f(x)=|\cos x|\) is a continuous function.
Examine that \(\sin|x|\) is a continuous function.
Find all the points of discontinuity of \(f\) defined by \(f(x)=|x|-|x+1|\).
There is no point of discontinuity.